The Geometry of Synchronization Problems and Learning Group Actions

The Geometry of Sync hronization Problems and Learning Group Actions Tingran Gao · Jacek Bro dzki · Sa yan Mukherjee Received: 16 April 2018 / Accepted: 29 April 2019 Abstract W e dev elop a geometric framework, based on the classical theory of ﬁbre bundles, to c harac- terize the cohomological nature of a large class of synchr onization-typ e pr oblems in the context of graph inference and combinatorial optimization. W e identify each synchronization problem in top ological group G on connected graph Γ with a ﬂat principal G -bundle ov er Γ , thus establishing a classiﬁcation result for sync hronization problems using the representation v ariet y of the fundamental group of Γ into G . W e then dev elop a twisted Ho dge theory on ﬂat vector bundles asso ciated with these ﬂat principal G -bundles, and pro vide a geometric realization of the gr aph c onne ction L aplacian as the low est-degree Ho dge Laplacian in the twisted de Rham-Ho dge co chain complex. Motiv ated by these geometric intuitions, we prop ose to study the problem of le arning gr oup actions — partitioning a collection of ob jects based on the local synchroniz- abilit y of pairwise corresp ondence relations — and provide a heuristic sync hronization-based algorithm for solving this type of problems. W e demonstrate the eﬃcacy of this algorithm on sim ulated and real datasets. Keyw ords synchronization problem · ﬁbre bundle · holonom y · Ho dge theory · graph connection Laplacian Mathematics Sub ject Classiﬁcation (2010) 05C50 · 62-07 · 57R22 · 58A14 1 Introduction Ov er the past century , concepts from diﬀerential geometry hav e had a strong impact on probabilit y theory , statistical inference, and machine learning [37, 48, 64, 108, 126]. Two central geometric concepts used in these ﬁelds ha v e b een diﬀeren tial operators (e.g. the Laplace–Beltrami operator [16]) and Riemannian metrics (e.g. Fisher information [64]). In particular, the researc h program of manifold learning studies dimension reduction through the lens of diﬀeren tial-geometric quantities and in v arian ts, and designs data compression algorithms TG gratefully ac kno wledges partial support from Simons Math+X In vestigators Award 400837, DARP A D15AP00109, NSF IIS 1546413, and an AMS-Simons T ra vel Grant; JB w ould lik e to ac knowledge the support for this w ork by the EPSRC grants EP/I016945/1 and EP/N014189/1; SM would like to ackno wledge supp ort from NSF DMS 16-13261, NSF IIS 1546331, NSF DMS-1418261, NSF I IS-1320357, NSF DMS-1045153, and HFSP RGP0051/2017. Tingran Gao Committee on Computational and Applied Mathematics, Department of Statistics, Universit y of Chicago, Chicago, IL 60637, USA E-mail: tingrangao@galton.uchicago.edu Jacek Bro dzki Department of Mathematical Sciences, Universit y of Southampton, Southampton SO17 1BJ, E-mail: j.bro dzki@soton.ac.uk Say an Mukherjee Departments of Statistical Science, Mathematics, Computer Science, and Bioinformatics & Biostatistics, Duke University , Durham, NC 27708, USA E-mail: say an@stat.duk e.edu 2 Tingran Gao, Jacek Bro dzki, and Say an Mukherjee that preserv e in trinsic geometric information suc h as geodesic distances [145], aﬃne connections [128], second fundamen tal forms [54], and heat kernels [15, 42, 43]. The underlying hypothesis of these tec hniques is that the data lie appro ximately on a smo oth manifold (often embedded in an am bien t Euclidean space), a scenario facilitating inference due to smo othly controllable transitions betw een observed and unseen data. F or practical purp oses, discrete analogues of the inheren tly smooth theory of diﬀerential geometry ha ve also been explored in ﬁelds ranging from geometry pro cessing [18, 49], ﬁnite elemen t methods [6], to sp ectral graph theory [38] and diﬀusion geometry [41, 136]. Bey ond the manifold assumption, geometric ob jects can b e handled with “softer” to ols such as top ology: top ological data analysis techniques [33, 57] hav e b een developed to study datasets based on their persistent homology . F or smo oth manifolds, it is w ell known that the singular cohomology and de Rham cohomology are isomorphic, indicating that some topological information can b e read oﬀ from the diﬀeren tial structure of geometric ob jects. Carrying the de Rham theory b eyond the manifold setting has attracted the interest of geometers and physicists: synthetic diﬀerential geometry [92, 93] deﬁnes group-v alued diﬀeren tial forms on “formal manifolds” (generalized notion of smo oth spaces for whic h inﬁnitesimal neighborho o ds are sp eciﬁed axiomatically), based on which an analog of the classical de Rham theory can b e established [63]; noncom- m utative diﬀeren tial geometry [45, 46, 106] builds up on the observ ation that muc h of diﬀerential geometry can b e formulated in terms of the algebra of smooth functions deﬁned on smo oth manifolds, and replaces this algebra with noncommutativ e ones — diﬀerential forms can then b e extended to “noncommutativ e spaces” along with homology and cohomology of m uc h more general ob jects. Discrete analogs of the Ho dge Laplacian, a second order diﬀeren tial operator closely related to de Rham theory , ha v e been proposed for sim- plicial complexes and graphs [85, 101, 121, 122, 140]; its non-comm utative coun terpart for 1-forms on graphs ha ve recently b een explored in [107]. Bridging recent developmen ts applying diﬀerential geometry and top ology in probability and statisti- cal sciences, the problem of synchr onization [13, 154] arise in a v ariet y of ﬁelds in computer science (e.g. computer vision [10] and geometry processing [91]), signal processing (e.g. sensor netw ork lo calization [51]), com binatorial optimization (e.g. non-comm utative Grothendiec k inequality [12]), and natural sciences (e.g. cry o-electron microscop y [11, 130, 138] and geometric morphometrics [66]). The data given in a sync hroniza- tion problem include a connected graph that enco des similarit y relations within a collection of ob jects, and pairwise corresp ondences — often realized as elements of a transformation group G — characterizing the nature of the similarit y betw een a pair of ob jects link ed directly b y an edge in the relation graph. The general goal of the problem is to adjust the pairwise corresp ondences, which often suﬀer from noisy or incomplete measuremen ts, to obtain a globally consisten t c haracterization of the pairwise relations for the en tire dataset, in the sense that un veiling the transformation b et ween a pair of ob jects far-apart in the relation graph can b e done by comp osing transformations along consecutiv e edges on a path connecting the t wo ob jects, and the resulting comp osed transformation is indep enden t of the c hoice of the path. (A precise deﬁnition of a sync hronization problem will b e provided b elo w; see Section 1.1.) This pap er stems from our attempt to gain a deep er understanding of the geometry underlying synchronization problems. Whereas previous works [136, 137] in this direction build up on manifold assumptions, the p oint of view w e adopt here is synthetic and noncomm utativ e: w e will see that inference is p ossible due to rigidit y rather than smoothness. The remainder of this section giv es a formal deﬁnition of sync hronization problems, as w ell as a geometric in terpretation in the language of ﬁbre bundles. The ﬁbre bundle in terpretation is elementary but has not b een presented in the literature of sync hronization problems, to our knowledge. W e then state the main results, discuss related w orks, and describ e the organization of the paper. 1.1 A Fibre Bundle Interpretation of Sync hronization Problems W e begin with a standard formulation of the sync hronization problem originated in a series of w orks b y A. Singer and collab orators [134, 130, 13, 154, 21, 11]. Let Γ = ( V , E , w ) b e an undirected weigh ted graph with v ertex set V , edge set E , and w eights w ij for each ( i, j ) ∈ E . Assume G is a top ological group acting on a normed vector space F . Giv en a map ρ : E → G from the edges of Γ to the group G satisfying ρ ij = ρ − 1 j i , the ob jectiv e of a F -synchr onization pr oblem over Γ with r esp e ct to ρ is to ﬁnd a map f : V → F satisfying the constrain ts f i = ρ ij f j ∀ ( i, j ) ∈ E . (1) The Geometry of Synchronization Problems and Learning Group Actions 3 T able 1 Notations used throughout this pap er Γ graph V vertex set of Γ E edge set of Γ n or | V | num ber of vertices of the graph Γ m or | E | num ber of edges of the graph Γ w ij weigh t on edge ( i, j ) ∈ E d i weigh ted degree on vertex i ∈ V , deﬁned as d i = P j :( i,j ) ∈ E w ij G topological group e identit y elemen t of G G δ group G equipp ed with discrete top ology K scalar ﬁeld R or C F v ector space on K that is a representation space of G d or dim F dimension of the vector space F h· , ·i F inner pro duct on F U = { U i | 1 ≤ i ≤ | V |} open cov er of Γ in which U i is the star of vertex i ∈ V C 0 ( Γ ; G ) G -v alued 0-cochain on Γ , or the set of all vertex p otentials on Γ C 1 ( Γ ; G ) G -v alued 1-cochain on Γ , or the set of all edge p otentials on Γ C 0 ( Γ ; F ) F -v alued 0-co chain on Γ B ρ synchronization principal bundle (a ﬂat principal G -bundle on Γ ) asso ciated with ρ ∈ C 1 ( Γ ; G ) B ρ [ F ] ﬂat asso ciated F -bundle of B ρ hol ρ holonomy homomorphism on B ρ , from π 1 ( Γ ) to G Hol ρ ( Γ ) holonomy of the synchronization principal bundle B ρ Ω 0 i ( Γ ; B ρ [ F ]) constant twisted lo cal 0-forms of B ρ [ F ] on U i , i.e. constant lo cal sections of B ρ [ F ] on U i Ω 0 ( Γ ; B ρ [ F ]) locally constan t t wisted global 0-forms of B ρ [ F ], i.e. locally constan t global sections of B ρ [ F ] Ω 1 i ( Γ ; B ρ [ F ]) constant twisted lo cal 1-forms of B ρ [ F ] on U i Ω 1 ( Γ ; B ρ [ F ]) locally constant twisted global 1-forms of B ρ [ F ] [ f ] vector in K nd representing f ∈ C 0 ( Γ ; F ) d ρ ρ -twisted diﬀerential where ρ ∈ C 1 ( Γ ; G ), from C 0 ( Γ ; F ) to Ω 1 ( Γ ; B ρ [ F ]) δ ρ ρ -twisted co diﬀerential where ρ ∈ C 1 ( Γ ; G ), from Ω 1 ( Γ ; B ρ [ F ]) to C 0 ( Γ ; F ) ∆ (0) ρ ρ -twisted Ho dge Laplacian of degree 0 ∆ (1) ρ ρ -twisted Ho dge Laplacian of degree 1 H 0 ρ ( Γ ; B ρ [ F ]) The 0th twisted cohomology group for B ρ [ F ], where ρ ∈ C 1 ( Γ ; G ) ν ( S ) frustration of the subgraph of Γ spanned by the vertex subset S ⊂ V If no suc h map f exists, the synchronization problem consists of ﬁnding a map f from V to F that satisﬁes the constrain ts as m uch as possible, in the sense of minimizing the frustr ation η ( f ) = 1 2 X i,j ∈ V w ij k f i − ρ ij f j k 2 F X i ∈ V d i k f i k 2 F , (2) where k·k F is a norm deﬁned on F , and d i = P j :( i,j ) ∈ E w ij is the weigh ted degree at vertex i . In the terminology of [13], ρ is an e dge p otential and f is a vertex p otential ; a vertex potential is said to satisfy a giv en edge p otential if all equalities in (1) hold. V arying the choice of group G and ﬁeld F results in diﬀerent realizations of the sync hronization problem [135, 138, 147, 130, 154, 13, 21], as will b e elaborated in Section 1.3. Since we will frequen tly refer to the set of all edge and vertex p oten tials on a graph, let us in tro duce the follo wing notations to ease our exp osition: let C 0 ( Γ ; G ), C 1 ( Γ ; G ) denote respectively the set of all G -v alued v ertex and edge p oten tials on Γ , i.e. C 0 ( Γ ; G ) := { f : V → G } , C 1 ( Γ ; G ) :=  ρ : E → G | ρ ij = ρ − 1 j i , ∀ ( i, j ) ∈ E  . (3) F or cohomological reasons that will b ecome clear in Section 2, we will also call C 0 ( Γ ; G ) and C 1 ( Γ ; G ) the G -value d 0 - and 1 -c o chains on Γ , resp ectiv ely . Similarly , let C 0 ( Γ ; F ) := { f : V → F } (4) denote the set of all F -v alued vertex potentials on Γ . Throughout this pap er, a G -v alued edge p otential ρ ∈ Ω 0 ( Γ ; G ) is said to b e synchr onizable if there exists a G -v alued v ertex p otential f ∈ Ω 0 ( Γ ; G ) satisfying 4 Tingran Gao, Jacek Bro dzki, and Say an Mukherjee f i = ρ ij f j , ∀ ( i, j ) ∈ E , i.e. (1) is satisﬁed with F = G . Generally , an F -v alued vertex p oten tial satisfying (1) will b e referred to as a solution to the F -synchr onizable pr oblem over Γ with r esp e ct to ρ , or simply F -value d synchr onization solution . Clearly , ρ is synchronizable if and only if a G -v alued synchronization solution exists. When F = G , i.e. when we consider the action of G on itself, a sync hronizable edge potential can b e realized geometrically as a ﬂat 1 principal bundle that is isomorphic to a product space in its en tirety , i.e. a trivial 2 ﬂat principal bundle, as will b e explained in Prop osition 1.1 and Prop osition 1.2 b elow; this observ ation forms the backbone of the entire geometric framew ork we develop in this pap er. When the ﬁbre bundle is diﬀeren tiable, this notion of ﬂatness is equiv alent to the existence of a ﬂat connection on the bundle, whic h is essentially a special case of the Riemann-Hilb ert corresp ondence [59]. The main results of this pap er build up on extending further and deeper the analogy b etw een the geometry of synchronization problems and ﬁbre bundles. Prop osition 1.1 and Prop osition 1.2 characterize the basic building blo ck for the geometric formulation of synchronization problems. W e will develop the principal bundle in the generalit y of top ological spaces that includes smo oth structures as particular cases. F ollo wing Steenro d [141], a ﬁbre bundle is a quintuple E = ( E , M , F, π , G ) where E , M , F are top ological spaces, referred to as the total sp ac e , b ase sp ac e , and ﬁbr e sp ac e , resp ectively; π : E → M is a contin uous surjectiv e map, called the bund le pr oje ction , and M adopts an op en cov er { U i } with homeomorphisms φ i : U i × F → π − 1 ( U i ) b etw een each π − 1 ( U i ) ⊂ E and the pro duct space U i × F , such that π   π − 1 ( U i ) is the comp osition of φ i with pro j 1 : U i × F → U i , the canonical pro jection on to the ﬁrst factor of the product space. In other words, the follo wing diagram is comm utative: π − 1 ( U i ) U i × F U i φ i π Pro j 1 The op en co ver { U i } and the homeomorphisms { φ i } together pro vides a system of lo c al trivializations for the ﬁbre bundle E . Moreov er, G is a top ological transformation group on F enco ding the compatibility of “c hange-of-co ordinates” on M , with resp ect to the provided lo cal trivializations, in the following sense: at ev ery x ∈ U i ∩ U j 6 = ∅ , the restriction of the comp osed map φ − 1 i ◦ φ j : U j × F → U i × F on { x } × F , whic h necessarily gives rise to a homeomorphism from { x } × F to itself by deﬁnition, is canonically identiﬁed with a group element g ij ( x ) ∈ G , and the map g ij : U i ∩ U j → G is contin uous. The topological group G is called the structur e gr oup of the ﬁbre bundle E . The notation F x is often used to denote π − 1 ( x ) for x ∈ M , and referred to as the ﬁbr e over x ∈ M . It is straigh tforward to c hec k from these deﬁnitions that g ii ( x ) = e ∀ x ∈ U i (5) g ij ( x ) = g − 1 j i ( x ) ∀ x ∈ U i ∩ U j (6) g ij ( x ) g j k ( x ) = g ik ( x ) ∀ x ∈ U i ∩ U j ∩ U k (7) where e is the iden tity elemen t of the structural group G . The family of con tin uous maps { g ij : U i ∩ U j → G } is called a system of c o or dinate tr ansformations for the ﬁbr e bund le E . Interestingly , essentially all information for determining the ﬁbre bundle E is enco ded in the co ordinate transformations, as the follo wing theorem indicates: Theorem 1.1 (Steenro d [141] § 3.2) If G is a top olo gic al tr ansformation gr oup of F , U j is an op en c over of M , { g ij } is a family of c ontinuous maps fr om e ach non-empty interse ction U i ∩ U j to G satisfying (5) , (6) , (7) , then ther e exists a ﬁbr e bund le E with b ase sp ac e M , ﬁbr e F , structur al gr oup G , and c o or dinate tr ansformations { g ij } . Any two such ﬁbr e bund les ar e e quivalent to e ach other. The precise deﬁnition for tw o ﬁbre bundles with the same base space, ﬁbre space, and structural group to b e equiv alen t can b e found in [141, § 2.4], but w e will also cov er it in Section 2.1. Notice that the conditions 1 Recall (see, e.g. [146, § 2]) that a ﬁbre bundle π : B → X , with total space B and base space X , is said to b e ﬂat if it admits a system of lo cal trivializations with lo cally constant bundle co ordinate transformations. 2 Note that a ﬂat bundle is not necessarily trivial (i.e. isomorphic to a pro duct space) — the fundamental group of the base space plays a central role in this developmen t (see e.g. [116, Chapter 2]). The Geometry of Synchronization Problems and Learning Group Actions 5 (5), (6), (7) are reminiscent of the characterization for the synchronizabilit y (1) of a G -v alued edge p oten tial on a connected graph: if ρ satisﬁes (1) for a map f : V → G , then ρ ij = f i f − 1 j on each edge ( i, j ) ∈ E , which certainly satisﬁes ρ ii = e ∀ i ∈ V , ρ ij = ρ − 1 j i ∀ ( i, j ) ∈ E , ρ ij ρ j k = ρ ik ∀ ( i, j ) , ( j, k ) , ( i, k ) ∈ E . (8) As the following Proposition 1.1 establishes, viewing the graph Γ as a top ological space, an appropriate op en cov er of Γ can be found such that any synchronizable edge p oten tial can b e realized as co ordinate transformations of a ﬁbre bundle with base space Γ and the top ological group G serving b oth as the ﬁbre space and the structure group. A ﬁbre bundle with its structural group as ﬁbre t yp e is called a princip al bund le . Moreov er, any suc h principal bundle m ust also b e ﬂat, as the bundle co ordinate transformations tak e constan t v alues on every non-empty intersection of sets in the op en co ver. The following simple concepts from com binatorial graph theory and algebraic topology (see, e.g. [14, 20]) will be needed for the statement and proof of Prop osition 1.1: 1) The n -skeleton of a simplicial complex K is the subcomplex of K consisting of all j -dimensional faces for 0 ≤ j ≤ n ; 2) The supp ort of a simplicial complex K is the underlying top ological space of K ; 3) The star neighb orho o d of a vertex v in a simplicial complex K is the union of all closed simplices in K con taining v as a vertex; 4) A clique c omplex of a graph Γ = ( V , E ) is the simplicial complex with all complete subgraphs of Γ as its faces. Prop osition 1.1 L et G b e a top olo gic al gr oup, Γ = ( V , E ) a c onne cte d undir e cte d gr aph, and ρ : E → G a map satisfying ρ ij = ρ − 1 j i for al l ( i, j ) ∈ E . Denote X for the 2 -skeleton of the clique c omplex of the gr aph Γ , X the supp ort of X , and U = { U i | 1 ≤ i ≤ | V |} for an op en c over of X in which U i is the interior of the star of vertex i . Then ρ is synchr onizable over G if and only if ther e exists a ﬂat trivial princip al ﬁbr e bund le π : P ρ → X with structur e gr oup G and a system of lo c al trivializations deﬁne d on the op en sets in U with c onstant bund le tr ansition functions ρ ij on non-empty U i ∩ U j . A pro of of Prop osition 1.1 can b e found in App endix A. The key idea is to view Γ as the 1-skeleton of its asso ciated clique complex, and use the op en cov er consisting of star neighborho o ds of eac h vertex. A similar construction of “Cryo-EM complex” has b een used in [158] to classify data input to Cry o-EM problems, an imp ortan t application of sync hronization tec hniques. Ho wev er, it is imp ortan t to notice that the con verse to Prop osition (1.1) is not true in general; more precisely , an edge p otential satisfying (8), which necessarily sp eciﬁes a ﬂat principal bundle ov er Γ , need not b e synchronizable. F or a simple example, consider a square graph Γ consisting of a four v ertices 1, 2, 3, 4 and four edges (1 , 2), (2 , 3), (3 , 4), (4 , 1), forming a closed simple lo op but without any triangles enclosed b y three edges. An edge p otential satisfying ρ ij = ρ − 1 j i on all edges clearly satisﬁes all equalities in (8) since no consistency needs to b e chec k ed on edge triplets, but it is easy to ﬁnd ρ violating the equality ρ 12 ρ 23 ρ 34 ρ 41 = e which must be obeyed b y any synchronizable edge p otential, pro vided that the group G is not trivial. The lesson is that the compatibilit y conditions (8) are of a lo cal nature, in the sense that the cycle-c onsistency (b orrowing a term from geometry processing of shape collections [119, 82] that describes a compatibility constraint analogous to the last equalit y in (8)) is imp osed only on triangles comp osed of edge triplets; in contrast, synchronizabilit y requires a stronger notion of “global” cycle-consistency for the op eration of comp osing group elemen ts along loops of arbitrary length and topology on he graph. In a certain sense, ﬁbre bundles are the geometric mo dels realizing edge p oten tials that are “locally synchronizable.” Prop osition 1.1 is our ﬁrst attempt at understanding the geometric mechanism of sync hronization prob- lems. The assumption of the synchronizabilit y of ρ signiﬁcan tly restricts the range of applicabilit y of this geometric analogy: in most scenarios of interest, the synchronizabilit y of an edge p otential is the goal rather than the starting point for a sync hronization problem. F ortunately , it is p ossible to extend the ﬁbre bundle analogy b ey ond the sync hronizabilit y assumption in Proposition 1.1, b y restricting the model base space from the 2-skeleton of the clique complex of the graph to the 1-skeleton, and adjust the op en cov er U accordingly: if w e deﬁne an op en co ver U on the graph Γ (which as a topological space is canonically identiﬁed with the 1-sk eleton of its clique complex) in whic h eac h open set U i co vers only v ertex i and the in terior of all edges adjacen t to it, then U i ∩ U j 6 = ∅ if and only if ( i, j ) ∈ E , and any triple in tersection of op en sets in U is 6 Tingran Gao, Jacek Bro dzki, and Say an Mukherjee empt y . In consequence, any system of bundle co ordinate transformations deﬁned on U by a G -v alued edge p oten tial ρ automatically satisﬁes (5), (6), (7), and sp eciﬁes a ﬂat principal G -bundle ov er Γ , denoted as B ρ , regardless of synchronizabilit y . This is also consisten t with the deﬁnition of v ector bundles on graphs in [90]. Clearly , when ρ is sync hronizable, B ρ is the restriction of the principal G -bundle P ρ in Prop osition 1.1 to the 1-sk eleton of the base space Γ , therefore trivial as w ell. Con versely , if B ρ is trivial, b y [141, § 2.10 or § 4.3], there exists a map f : Γ → G assigning a constant v alue f i for all p oints x ∈ U i suc h that ρ ij = f i f − 1 j for all U i ∩ U j 6 = ∅ , which gives rise to a map f : V → G by restriction to the vertex set V of Γ ; this veriﬁes all constrain ts in (1) and establishes the synchronizabilit y of the edge p oten tial ρ . Consequently , the trivialit y of B ρ and P ρ implies each other, both are equiv alent to the synchronizabilit y of ρ . W e summarize these observ ations in Proposition 1.2 and formally deﬁne the synchr onization princip al bund le B ρ , whic h will b e of cen tral imp ortance for the geometric framew ork we develop in the rest of this pap er. Prop osition 1.2 L et G b e a top olo gic al gr oup, Γ = ( V , E ) a c onne cte d undir e cte d gr aph, and ρ : E → G a map satisfying ρ ij = ρ − 1 j i for al l ( i, j ) ∈ E . Write U = { U i | 1 ≤ i ≤ | V |} for an op en c over of Γ in which U i is the union of the single vertex set { i } with the interior of al l e dges adjac ent to the vertex i . Then ρ deﬁnes a ﬂat princip al G -bund le B ρ over Γ with a system of lo c al trivializations deﬁne d on the op en sets in U with c onstant bund le tr ansition functions ρ ij on non-empty U i ∩ U j . F urthermor e, ρ is synchr onizable if and only if B ρ is trivial. Deﬁnition 1.1 (Sync hronization Principal Bundle) The ﬁbre bundle B ρ asso ciated with the connected graph Γ and edge p otential ρ as characterized in Prop osition 1.2 is called a synchr onization princip al bund le of e dge p otential ρ over Γ , or a synchr onization princip al bund le for short. In practice, it is often more con venien t to w ork with B ρ rather than P ρ , not only since non-synchronizable edge p oten tials are muc h more prev alent, but also b ecause noisy or incomplete measurements almost alwa ys cause the observed group elements ρ ij ∈ G to b e non-sync hronizable. Solving for a G -v alued sync hronization solution can thus b e viewed as an approach to “denoising” or “ﬁltering” those observed transformations ρ ij , as was already implicit in many applications [13, 12]. In the sense of Proposition 1.2, these problems can b e in terpreted as inference on the structure of ﬂat principal bundles. Most sync hronization problems in practice [135, 138, 130, 147] considers vertex p otentials v alued in G , the same top ological group in whic h the prescrib ed edge p otential takes v alue, p ertaining to the principal bundle picture (i.e. F = G ) discussed in Prop osition 1.1 and Prop osition 1.2. Our ﬁbre bundle interpretation naturally includes more general synchronization problems in which the vertex p oten tials tak es v alues in F 6 = G as well, by relating the synchronizabilit y of an edge p otential to the existence of global sections on an asso ciate d F -bund le P ρ × η F or B ρ × η F , where η : G × F → F denotes the action of G on F . Essen tially , an asso ciated F -bundle P ρ × η F (or B ρ × η F ) is constructed using the same pro cedure as the principal bundle P ρ (or B ρ ), but with ﬁbre F instead of G . The associated bundles P ρ × η F , B ρ × η F are thus also ﬂat (but not necessarily trivial), since their bundle co ordinate transformations are equiv alen t to those of their principal bundle, up to the represen tation induced by the action η . A ma jor diﬀerence b etw een working with an asso ciated bundle and the principal bundle is that the co cycle condition ρ kj ρ j i = ρ ki ma y still not b e satisﬁed in the presence of a v ertex potential f : V → F satisfying (1), as elements in F can not b e “inv erted” in general; another imp ortant diﬀerence is the relation b et ween trivialit y and global sections: whereas a principal bundle P ρ or B ρ is trivial if and only if one global section exists, which amounts to ﬁnding one solution to the synchronization problem ov er Γ with resp ect to ρ , an asso ciated bundle may admit one or more global sections y et still b e non-trivial. F or instance, a vector bundle alwa ys admits the zero global section, regardless of its triviality . It turns out that establishing the sync hronizability of an edge p oten tial through the triviality of an asso ciated bundle requires ﬁnding “suﬃcien tly many” global sections of an asso ciated bundle, or in terms of synchronization problems, “suﬃcien tly man y” solutions satisfying (1). This is also reﬂected in the twisted Ho dge theory w e dev elop in Section 2.2. Ev en though ﬁnding enough global sections seems to b e more w ork, in practice it could b e m uch easier to ﬁnd a set of global sections on the asso ciated bundle than to ﬁnd ev en only one global section on the principal bundle, as the action of G on the space F introduces additional information from both geometric and practical p oin ts of view. Since the ﬂat F -bundle associated with B ρ will be essen tial for Section 2.2, w e in troduce the following deﬁnition: The Geometry of Synchronization Problems and Learning Group Actions 7 Deﬁnition 1.2 (Sync hronization Asso ciated Bundle) The ﬂat F -bundle on Γ asso ciated with the ﬂat principal bundle B ρ b y the action of G on F is called a synchr onization asso ciate d F -bund le of e dge p otential ρ over Γ , or synchr onization asso ciate d bund le for short, denoted as B ρ [ F ]. W e close this preliminary section dra wing analogy b etw een synchronization problems and ﬁbre bundles b y clarifying the relation among, and the geometric implications of, some v arian ts of the optimization for- m ulation of sync hronization problems. Giv en a graph Γ = ( V , E ) and a G -v alued edge p otential ρ , a direct translation of the goal of ﬁnding an F -v alued vertex p oten tial f satisfying (1) as muc h as p ossible is to solve min f : V → F X ( i,j ) ∈ E Cost F ( ρ ij f j , f i ) , (9) where Cost F : F × F → [0 , ∞ ) is a cost function on F (e.g., derived from a distance or a norm). When w e seek multiple solutions to an F -sync hronization problem o ver Γ with prescrib ed edge p otential ρ , it is natural to imp ose the additional constraints that the solutions should b e suﬃciently diﬀerent from each other; in the presence of a Hilb ert space structure on F , it is con venien t to imp ose orthogonality constraints b et ween pairs of solutions to obtain linearly indep endence. With additional normalization constraints to ﬁx the issue of identiﬁabilit y , this exactly translates in to the sp ectral relaxation algorithm in [13]. If the prescrib ed edge p otential is sync hronizable, its synchronizabilit y will b e conﬁrmed once a suﬃcien t num b er of synchronization solutions can b e collected, where the actual num b er dep ends on the prop erty of the group G as well as its action on F ; if not, a sync hronizable edge potential can b e constructed from suﬃcien tly man y F -v alued “approximate solutions” that minimize the ob jectiv e function in (9) as muc h as p ossible. The case G = F corresp onds to the optimization problem min f : V → G X ( i,j ) ∈ E Cost G ( ρ ij f j , f i ) . (10) whic h, in the case the Cost G is G -in v ariant, is equiv alen t to min f : V → G X ( i,j ) ∈ E Cost G  ρ ij , f i f − 1 j  . (11) If ρ is synchronizable, a minimizer of (10) (resp. (9)) attaining zero ob jectiv e v alue can b e geometrically realized as a global section of the sync hronization principal bundle B ρ (resp. P ρ ); such a minimizer implies the triviality of the principal bundle B ρ (resp. P ρ ), but not necessarily so in general for the asso ciated bundle B ρ × η F (resp. P ρ × η F ). If ρ is not sync hronizable, the minimum v alues of (10), (11), and (9) are all greater than zero, and minimizer of (10) or (11) can be view ed as a “denoised” or “ﬁltered” version of a trivial ﬂat principal bundle underlying the dataset. 1.2 Main Contributions In this section we give a brief o verview of our main contribution. W e will motiv ate the tw o ingredients of the geometric framework developed in Section 2, namely , holonomy representation and Ho dge theory , b y demonstrating preliminary versions of our formulation that lead to weak er conclusions or incomplete geometric pictures, then sk etch the full approaches adopted in Section 2. Finally , w e draw the link betw een the geometric framework and the prop osal of the le arning gr oup actions (LGA) problem. 1.2.1 Holonomy of Synchr onization Princip al Bund les Consider B ρ , the sync hronization principal bundle arising from a G -sync hronization problem ov er a con- nected graph Γ = ( V , E ) with resp ect to ρ ∈ C 1 ( Γ ; G ). Fix an arbitrary vertex v ∈ V , and denote the set of all v -based lo ops in Γ (lo ops with v as b oth the starting and ending v ertex) as Ω v ; Ω v carries a natural group structure by the composition of v -based loops. Now the procedure of taking the pro duct of the v alues of ρ along the consecutiv e edges in the lo op sp eciﬁes a group homomorphism from Ω v in to G . Denote the image of this group homomorphism by H v , whic h is necessarily a ﬁnitely generated subgroup of G since Γ 8 Tingran Gao, Jacek Bro dzki, and Say an Mukherjee is a ﬁnite graph. The simple but imp ortant observ ation here is that H v is the trivial subgroup of G if and only if ρ is synchronizable. The group H v is the analogy of the holonomy gr oup b ase d at v in diﬀerential geometry , if we view ρ ij on edge ( i, j ) as the p ar al lel-tr ansp ort b etw een ﬁbres of B ρ at i, j ∈ V . Section 2.1 is devoted to a deep er and more systematic treatment of the group homomorphism from lo ops in Γ to the structure group G . W e will deﬁne Hol ρ ( Γ ), the holonomy of the synchronization prin- cipal bundle B ρ (indep enden t of the choice of the base vertex v ), as well as an equiv alence relation on C 1 ( Γ ; G ) induced b y a right action of C 0 ( Γ ; G ) (which is treated implicitly when solving synchroniza- tion problems in practice), and establish a corresp ondence b etw een Hol ρ ( Γ ) and the equiv alence class in C 1 ( Γ ; G ) /C 0 ( Γ ; G ) to whic h ρ ∈ C 1 ( Γ ; G ) b elongs. In particular, trivial holonomy Hol ρ ( Γ ) corresp onds to the orbit in C 1 ( Γ ; G ) /C 0 ( Γ ; G ) consisting precisely of all synchronizable edge p oten tials. This corre- sp ondence will be form ulated in Theorem 2.1 as betw een C 1 ( Γ ; G ) /C 0 ( Γ ; G ) and Hom ( π 1 ( Γ ) , G ) /G , the r epr esentation variety of the fundamental group of Γ into G . 1.2.2 Twiste d De Rham Cohomolo gy of Synchr onization Asso ciate d V e ctor Bund les The gr aph c onne ction L aplacian (GCL) for an F -sync hronization problem ov er graph Γ with resp ect to ρ ∈ Ω 1 ( Γ ; G ) is a linear operator on Ω 0 ( Γ ; F ) deﬁned as ( L 1 f ) i := 1 d i X j :( i,j ) ∈ E w ij ( f i − ρ ij f j ) , ∀ i ∈ V , ∀ f ∈ C 0 ( Γ ; F ) . If F is a v ector space and G has a matrix represen tation on F , GCL can b e written as a block matrix in whic h the ( i, j )-th blo ck is the matrix representation of ρ ij , if ( i, j ) ∈ E . GCL essentially carries all information of a sync hronization problem and is of cen tral imp ortance to the sp ectral and SDP relaxation algorithms for synchronization. Our motiv ation for Section 2.2 w as to provide a cohomological interpretation for GCL, in the hop e of realizing it geometrically as a Ho dge Laplacian in a co chain complex, inspired by a similar geometric realization of the gr aph L aplacian (in the con text of algebraic and sp ectral graph theory) as a Ho dge Laplacian on degree-zero forms in discrete Ho dge theory (see App endix B). Note that GCL reduces to the graph Laplacian if the group G is a scalar ﬁeld. In the literature of diﬀerential geometry , twiste d diﬀer ential forms on a ﬂat vector bundle E can b e in tuitively thought as bundle-v alued diﬀerential forms on the base manifold. The twisted Ho dge theory w e develop in Section 2.2 deﬁnes t wo discrete diﬀerential op erators that are formal adjoints of each other b et ween c onstant twiste d lo c al 0 -forms and c onstant twiste d lo c al 1 -forms on the synchronization asso ciated F -bundle B ρ [ F ] , namely the ρ -twiste d diﬀer ential d ρ and the ρ -twiste d c o diﬀer ential δ ρ , suc h that GCL can b e written as the comp osition δ ρ d ρ . Provisionally , by identifying each f ∈ C 0 ( Γ ; F ) naturally with a collection of constant t wisted lo cal 0-forms — one for each op en set U i ∈ U — a coarse approximation of our construction can b e written as ( d ρ f ) ij ∼ f i − ρ ij f j , ∀ f ∈ C 0 ( Γ ; F ) , ( δ ρ ω ) i ∼ 1 d i X j :( i,j ) ∈ E w ij ω ij , ∀ ω ∈ C 1 ( Γ ; F ) := { ω : E → F | ω ij = − ω j i ∀ ( i, j ) ∈ E } , from which it can be easily chec k ed that L 1 = δ ρ d ρ on C 0 ( Γ ; F ). The main conceptual diﬃculty with this natural form ulation is that “ d ρ f ” deﬁned as such does not p ossess the skew-symmetry desired for 1-forms, since in general f j − ρ j i f j = − ρ j i ( f i − ρ ij f j ) 6 = − ( f i − ρ ij f j ) . (12) The framework we dev elop in Section 2.2 circumv ent this skew-symmetry issue with 1-forms by deﬁning f i − ρ ij f j as the representation of d ρ f , a twiste d glob al 1 -form deﬁned ov er the entire graph Γ , in the system of lo cal trivializations of B ρ [ F ] ov er the op en co v er U . W e then deﬁne the ρ -twiste d c o diﬀer ential δ ρ that is the formal adjoint of d ρ with resp ect to inner pro ducts naturally sp eciﬁed on the space of twisted lo cal 0- and 1-forms, and realize the graph connection Laplacian L 1 as the degree-zero Ho dge Laplacian δ ρ d ρ : C 0 ( Γ ; G ) → C 0 ( Γ ; G ) in the twisted de Rham co chain complex (41). These constructions lead to t wo diﬀerent characterizations of the synchronizabilit y of ρ , one in Prop osition 2.3 with a t wisted de Rham cohomology group, and the other through a Hodge-type decomposition of C 0 ( Γ ; F ) follo wing Theorem 2.2. The Geometry of Synchronization Problems and Learning Group Actions 9 This twisted Ho dge theory also provides geometric insigh ts for the GCL-based sp ectral relaxation algorithm and Cheeger-t ype inequalities in [13], as we will elaborate in Section 2.2.4. 1.2.3 L e arning Gr oup A ctions via Synchr onizability Fibre bundles are top ological spaces that are pro duct spaces lo cally but not necessarily globally . How ever, w e can still lo ok for maximal op en subsets of the base space on which the ﬁbre bundle is trivializable, and seek a decomp osition of the base space into the union of such “maximal trivializable subsets.” This in tuition motiv ated us to consider applying synchronization tec hniques to partition a graph in to connected comp onen ts, based on the synchronizabilit y of a prescrib ed edge p oten tial in addition to the connectivity of the graph. In Section 3.1, we deﬁne the general problem of le arning gr oup actions (LGA) for a set X , equipp ed with an action by group G , as searching for a partition of X into a sp eciﬁed num b er of subsets and learning a new group action on X that is cycle-consisten t within eac h partition; the cycle-consistency need not b e maintained for a cycle of actions across multiple partitions. The LGA problem is then sp ecialized to the setting of synchronization problems ( le arning gr oup actions by synchr onization , or LGAS), for which w e deﬁne a quantit y that measures the p erformance of graph partitions based on the synchronizabilit y of a ﬁxed edge p otential on the entire graph Γ , motiv ated b y the classical normalized graph cut algorithm. Finally , w e propose in Section 3.2 a heuristic algorithm for LGAS, building upon iteratively applying existing sync hronization tec hniques hierarc hically and p erforming spectral clustering on the edge-wise frustration. 1.3 Broader Context and Related W ork The sync hronization problem has b een studied for a v ariet y of choices of top ological groups G and spaces F . T ypically these formulations fall into our principal bundle setting and require an underlying manifold structure. W e give a brief summary for the corresp ondences b et ween c hoices of G and F in our framework and the practical instances in the sync hronization literature: In [13] G = F = O ( d ) and G = O ( d ), F = S d − 1 are studied; the case G = F = SO ( d ) is examined in [21, 154]; orientation detection or G = F = O (1) is considered in [135]; cryo-electron microscop y concerns G = F = SO (2) [130, 138]; globally aligning three- dimensional scans is the case where G = F = SO (3), and so is [147]. Our formulation of the synchronization problem considers a broader class of geometric structure than what has b een prop osed in the literature. Sp eciﬁcally , we do not require a manifold assumption (as the problem is mo deled on top ological spaces), or a principal bundle structure (as we can work with any asso ciated bundle), or compact and/or commutativ e structure groups. F or comparison, the vector and principal bundle framew ork developed in [136, 137] relies on manifold assumptions for the base, ﬁbre, and total space, as w ell as an (extrinsic) isometric em b edding into an ambien t Euclidean space for lo cally estimating tangent spaces and parallel-transports; similarly for recen t w ork [67] extending this geometric framework to smo oth bundles with general ﬁbre types. Both V ector Diﬀusion Maps (VDM) [136] and Horizontal Diﬀusion Maps (HDM) [67] can b e viewed as attempts at combining the idea of synchronization with diﬀusion geometry [41, 42, 43]. The geometry underlying the synchronization problem related to cryo-electron microscopy [130, 138] can b e describ ed using the language of SO (2)-principal bundles, as recen tly demonstrated in [158], with a ˇ Cec h cohomology approach through Leray’s Theorem whic h dep ends essentially on the commutativit y of the structure group SO (2), whereas most sync hronization problems of practical in terest in v olve noncommutativ e structure groups. The Non-Unique Games (NUG) and SDP relaxation framework established in [11, 10] assumes the compactness of the structure group G , and resorts to a compactiﬁcation pro cedure that maps a subset of G to another compact group for sync hronization problems ov er non-compact groups such as the Euclidean group in the motion estimation problem in computer vision [110, 78]. The graph t wisted Ho dge theory w e develop in Section 2.2 also has ties to recent developmen ts in discrete Ho dge theory [85, 101, 117, 121, 122, 140]. In [90] Laplacians on one- and tw o-dimensional vector bundles on graphs were used to understand the relation b etw een graphs embedded on surfaces and cycle ro oted spanning forests, generalizing the relation b etw een spanning trees and graph Laplacians. V arian ts of the graph Laplacian ha ve b een used to relate ranking problems to synchronization problems in [50, 62]; a com binatorial Laplacians based on a discrete Ho dge theory on directed graphs has been successfully applied to decomp ose ranking problems and games into “gradien t-like” v ersus “cyclic” comp onents in [85, 32], and 10 Tingran Gao, Jacek Bro dzki, and Say an Mukherjee to visualize directed netw orks in [61]. Discrete Laplacians on simplicial complexes hav e been prop osed, and sp ectral prop erties such as Cheeger inequalities and stationary distributions of random walks hav e b een examined in a series of pap ers [85, 101, 117, 121, 122, 140]. The co cycle conditions (8) are also characterized in geometry pro cessing and computer vision recently for analysis of shap e or image collections [119, 82, 153, 83], where they are also kno wn as cycle-c onsistency c onditions . The geometric and top ological to ols we utilize in this pap er, namely those inv olving the top ology and geometry of ﬁbre bundles, are cov ered in most standard textb o oks, e.g. [141, 144, 20]. After Milnor’s semi- nal work on ﬂat connections on a Riemannian manifold [114], the relation b etw een ﬂat bundles and their holonom y homomorphisms b ecame widely known [87, 104, 71, 148, 47] and is still attracting interests of mo d- ern mathematical physicists (e.g. Higgs bundles and representation of the fundamental group [79, 131]). In a completely top ological setup, ﬂat bundles can b e c haracterized as ﬁbrations with a homotopy-in v arian t lifting prop erty (a top ological analogue of parallel-transp ort in diﬀerential geometry); essentially the same corresp ondence b etw een ﬂat bundles and holonomy homomorphisms is already known to Steenro d [141] and referred to as characteristic classes of ﬂat bundles [56, 104, 71, 59]. In a broader con text, the corresp ondence b et ween ﬂat bundles (integrable connections) and lo cal systems (lo cally constant shea ves) is a sp ecial case of the Riemann-Hilb ert c orr esp ondenc e , a higher-dimensional generalization of Hilb ert’s tw en ty-ﬁrst problem [19, 5]. This corresp ondence fostered imp ortan t developmen ts in algebraic geometry , including D -mo dules [88, 89, 111, 112] and Deligne’s w ork on in tegrable algebraic connections [52]. Understanding the represen ta- tion v arieties of the fundamen tal groups of Riemann surfaces in to Lie groups has b een of interest to algebraic geometers, geometric top ologists, and representation theorists in the past decades [98, 72]. The rest of this pap er is organized as follows. Section 2 establishes the geometric framework for synchro- nization problems, relating the sync hronizability of an edge p otential ρ to (a) the trivialit y of the holonom y of the ﬂat principal bundle B ρ , in Section 2.1; (b) the dimension of the zero-th degree t wisted cohomology group of a ρ -twisted de Rham co chain complex, as w ell as the dimension of the k ernel of the zero-th de- gree twisted Ho dge Laplacian, in Section 2.2. Section 3 deﬁnes the problem of learning group actions, and prop oses SynCut, a heuristic algorithm based on synchronization and graph sp ectral tec hniques. Numerical sim ulations indicating the eﬀectiveness of SynCut is p erformed on syn thetic datasets in Section 3.3 and on a real dataset of a collection of anatomical surfaces in Section 4. A few problems of p otential interest are listed in Section 5 for future exploration. 2 Synchronization as a Cohomology Problem This section concerns t wo geometric aspects of the synchronization problem. Section 2.1 links the sync hro- nizabilit y of an edge p otential to the triviality of the holonomy group of a ﬂat principal bundle. Section 2.2 establishes a discrete twisted Ho dge theory that naturally realizes the graph connection Laplacian as the lo west-order Ho dge Laplacian of a t wisted de Rham cochain complex. The obstruction to sync hronizabilit y of an edge p otential turns out to b e a cohomology group in the twisted de Rham complex; the degeneracy of this cohomology group is reﬂected in the spectral information of the twisted Ho dge Laplacian, which also pro vides a geometric in terpretation for the relaxation techniques used in solving synchronization problems. 2.1 Holonom y and Sync hronizability The tw o main results in this section, Corollary 2.1 and Theorem 2.1, relate the synchronizabilit y of an edge p otential ρ to the trivialit y of the holonom y group of the synchronization principal bundle B ρ . Our motiv ation is as follo ws. Recall from Prop osition 1.2 that G -sync hronization problems on a ﬁxed graph Γ with diﬀerent edge p otentials are in one-to-one corresp ondence with ﬂat principal G -bundles ov er Γ , and the synchronizabilit y of an edge p otential translates into the triviality of the bundle; whereas the one-to-one corresp ondence is stated at the level of lo cal co ordinates in Prop osition 1.2, the triviality of the principal bundle is a prop erty of the equiv alence classes of ﬂat principal G-bundles, which suggests the same lev el of abstraction for synchronizabilit y . As will b e precisely stated later in this subsection, (appropriately deﬁned) equiv alence classes of edge potentials form the mo duli sp ac e of ﬂat G -bundles on Γ , and a giv en edge p otential is synchronizable if and only if it b elongs to the same equiv alence class as the trivial edge p otential that assigns eac h edge of Γ the iden tit y elemen t e ∈ G . The holonomy gr oup , or the equiv alence classes of holonomy The Geometry of Synchronization Problems and Learning Group Actions 11 homomorphisms from the fundamental group of Γ to the structure group G , is a faithful representation of the mo duli space of ﬂat principal G -bundles on Γ ; we thus will be able to detect the synchronizabilit y of an edge potential through the trivialit y of the associated holonomy group. This argumen t is reminiscen t of classical classiﬁcation theorems of (1) principal bundles with disconnected structure groups in topology (see e.g. [141, § 13.9]); (2) ﬂat connections in diﬀerential geometry (see e.g. [144, § 13.6]); and (3) holomorphic v ector bundles of ﬁxed rank and degree (see e.g. [156, Appendix § 2.1]) in complex geometry . F or f ∈ C 0 ( Γ ; G ), ρ ∈ C 1 ( Γ ; G ), we say that f and ρ ar e c omp atible on e dge ( i, j ) ∈ E if f i = ρ ij f j , and that f and ρ ar e c omp atible on gr aph Γ if they are compatible on every edge in Γ . Recall from Deﬁnition 1.1 that we write B ρ for the sync hronization principal bundle asso ciated with ρ ∈ C 1 ( Γ ; G ), as describ ed in Prop osition 1.2. Equiv alen tly , it is often conv enien t to view B ρ as a G δ -bundle on Γ , where G δ is the same group as G but equipp ed with the discrete top ology . F or ρ, ˜ ρ ∈ Γ , the G δ -bundles B ρ , B ˜ ρ are e quivalent , denoted as B ρ ∼ B ˜ ρ , if a bund le map (see [141, § 2.5]) exists betw een B ρ and B ˜ ρ that induces the identit y map on the base space Γ . Since B ρ , B ˜ ρ ha ve the same base space, ﬁbre, and structure group, recall from [141, Lemma 2.10] that they are equiv alen t if and only if there exist contin uous functions λ i : U i → G δ deﬁned on each U i ∈ U such that ˜ ρ ij = λ i ( x ) − 1 ρ ij λ j ( x ) , ∀ x ∈ U i ∩ U j 6 = ∅ . Since the top ology on G δ is discrete and U i is connected, λ i is constant on U i , and deﬁnes a vertex p oten tial b y setting f i := λ i ( v i ), where v i ∈ U i is the i th v ertex of Γ . This prov es the following lemma: Lemma 2.1 Two e dge p otentials ρ, ˜ ρ ∈ C 1 ( Γ ; G ) deﬁne e quivalent ﬂat princip al G -bund les on Γ if and only if ther e exists f ∈ C 0 ( Γ ; G ) such that ˜ ρ ij = f − 1 i ρ ij f j , ∀ ( i, j ) ∈ E . (13) In other wor ds, e quivalenc e classes of ﬂat princip al G -bund les on Γ determine d by e dge p otentials (thr ough Pr op osition 1.2) ar e in one-to-one c orr esp ondenc e with e quivalenc e classes in the orbit sp ac e C 1 ( Γ ; G ) /C 0 ( Γ ; G ) , wher e the right action of C 0 ( Γ ; G ) on C 1 ( Γ ; G ) is deﬁne d as [ f ( ρ )] ij := f − 1 i ρ ij f j , ∀ ( i, j ) ∈ E . (14) R emark 2.1 The orbit space C 1 ( Γ ; G ) /C 0 ( Γ ; G ) is exactly the ﬁrst c ohomolo gy set ˇ H 1 (( Γ , U ) , G ) for the sheaf of germs of lo cally constant G -v alued functions ov er Γ with resp ect to the op en cov er U , where G is p ossibly nonabelian. It is thus not surprising that the orbit space should iden tify naturally with isomorphism classes of ﬂat principal G -bundles o ver Γ (see e.g. [30, Proposition 4.1.2] or [113, § 8.1]). A p ath in Γ is a collection of consecutive edges in Γ . If all edges in path γ are orien ted consisten tly , w e sa y γ is an oriented path. F or an y oriented path γ , deﬁne γ − 1 the r everse of γ as the path in Γ consisting of the same consecutive edges in γ listed in the opp osite order and with all orientations reversed. F or an orien ted path γ consisting of consecutiv e edges { ( i 0 , i 1 ) , ( i 1 , i 2 ) , · · · , ( i N − 1 , i N ) } set hol ρ ( γ ) =  ρ i N ,i N − 1 ρ i N − 1 ,i N − 2 · · · ρ i 2 ,i 1 ρ i 1 ,i 0  − 1 = ρ i 0 , 1 ρ i 1 ,i 2 · · · ρ i N − 2 ,i N − 1 ρ i N − 1 ,i N ∈ G, (15) then hol ρ maps paths in Γ to elements of group G . F or tw o oriented paths γ , γ 0 suc h that the ending vertex of γ coincides with the starting vertex of γ 0 , deﬁne γ ◦ γ 0 as the orien ted path constructed b y concatenating γ 0 with γ . It is then straightforw ard to verify by deﬁnition that hol ρ  γ − 1  = hol ρ ( γ ) − 1 , hol ρ ( γ ◦ γ 0 ) = hol ρ ( γ ) hol ρ ( γ 0 ) . (16) If an orien ted path starts and ends at the same vertex v , we call it an oriente d lo op b ase d at vertex v . Denote Ω v for all lo ops based at v ∈ V in Γ , including the single vertex set { v } view ed as the identit y lo op based at v . Clearly , Ω v carries a group structure with the lo op concatenation and reversion op erations. The equalities in (16) ensures hol ρ ( { v } ) = e and that hol ρ : Ω v → G is a group homomorphism. Moreov er, since graph Γ do es not contain an y 2-simplices, tw o oriented lo ops based at v are homotopic if and only if they diﬀer b y a collection of disconnected trees in Γ , in whic h every tree gets mapp ed to e ∈ G under the map hol ρ ; the map hol ρ : Ω v → G thus descends naturally to a map to G from π 1 ( Γ , v ), the fundamental gr oup of Γ b ase d at v . Unless confusions arise, we shall also denote the descended map as hol ρ for simplicity of notation. Lemma 2.2 b elo w summarizes these discussions. 12 Tingran Gao, Jacek Bro dzki, and Say an Mukherjee Lemma 2.2 The map hol ρ : π 1 ( Γ , v ) → G deﬁne d in (15) is a gr oup homomorphism. In p articular, the image of this homomorphism is a sub gr oup of G . W e will refer to the group homomorphism hol ρ : π 1 ( Γ , v ) → G as the holonomy homomorphism at v ∈ Γ for a G -synchronization problem with prescrib ed edge p otential ρ ∈ C 1 ( Γ ; G ). Deﬁne the holonomy gr oup at v ∈ Γ of edge p otential ρ as the image Hol ρ ( v ) := hol ρ ( π 1 ( Γ , v )) . F rom a diﬀeren t p oint of view, Lemma 2.2 assigns an element of Hom ( π 1 ( Γ , v ) , G ) to each element of C 1 ( Γ ; G ), where Hom ( π 1 ( Γ , v ) , G ) is the set of group homomorphisms from π 1 ( Γ , v ) to G . Lemma 2.3 If Γ is c onne cte d, the holonomy gr oups Hol ρ ( v ) , Hol ρ ( w ) at v , w ∈ V ar e c onjugate to e ach other as sub gr oups of G . Pr o of Let γ b e a path in Γ connecting vertex v to vertex w . The fundamental groups of Γ based at v , w are related b y conjugation π 1 ( Γ , v ) = γ − 1 π 1 ( Γ , w ) γ , thus Hol ρ ( v ) = Hol ρ  γ − 1 π 1 ( Γ , w ) γ  = Hol ρ  γ − 1  Hol ρ ( π 1 ( Γ , w )) Hol ρ ( γ ) = Hol ρ ( γ ) − 1 Hol ρ ( w ) Hol ρ ( γ ) . Deﬁne the holonomy of ρ ∈ C 1 ( Γ ; G ) on a connected graph Γ as the follo wing conjugacy class (orbit of the action b y conjugation) of subgroups of G : Hol ρ ( Γ ) :=  g − 1 Hol ρ ( v ) g   for all g ∈ G, and an arbitrarily c hosen but ﬁxed v ertex v ∈ V  . (17) By Lemma 2.3, the deﬁnition of Hol ρ ( Γ ) is indep endent of the choice of a ﬁxed base v ∈ V . W e say that the holonom y of ρ ∈ C 1 ( Γ ; G ) is trivial on a connected graph Γ if Hol ρ ( Γ ) contains only the trivial subgroup { e } for all g ∈ G . Under the connectivit y assumption of Γ , the triviality of the global inv ariant Hol ρ ( Γ ) can b e completely determined by its seemingly “lo cal” counterparts; see Lemma 2.4 b elow. Of course, holonomy is not lo cal in nature, as Hol ρ ( v ) enco des the information of all oriented loops based at vertex v and in principle “touc hes” the en tire space Γ . Lemma 2.4 If Γ is c onne cte d, the fol lowing statements ar e e quivalent: (i) Hol ρ ( Γ ) is trivial; (ii) Hol ρ ( v ) = { e } for some vertex v ∈ V ; (iii) Hol ρ ( v ) = { e } for al l vertic es v ∈ V . Similar to the deﬁnition of Hol ρ ( Γ ) in (17), the fundamental group π 1 ( Γ ) of a connected graph Γ is also determined by the fundamen tal group π 1 ( Γ , v 0 ) at any v ertex v 0 ∈ V up to conjugacy classes. Therefore, Lemma 2.2 and Lemma 2.3 together assign to each ρ ∈ C 1 ( Γ ; G ) an equiv alence class in Hom ( π 1 ( Γ ) , G ) /G , in whic h G acts on Hom ( π 1 ( Γ ) , G ) b y the inner automorphisms of G φ 7→ g − 1 φg , ∀ φ ∈ Hom ( π 1 ( Γ ) , G ) , g ∈ G. In other w ords, Lemma 2.2 and Lemma 2.3 guaran tee a w ell-deﬁned map Hol : Ω 1 ( Γ ; G ) → Hom ( π 1 ( Γ ) , G ) /G . F urthermore, note in equation (15) that Hol is inv ariant under the righ t action (14) of C 0 ( Γ ; G ) on C 1 ( Γ ; G ), thus Hol naturally descends to a map from C 1 ( Γ ; G ) /C 0 ( Γ ; G ) to Hom ( π 1 ( Γ ) , G ) /G . The space Hom ( π 1 ( Γ ) , G ) /G is known as the r epr esentation variety of the fundamental group of Γ (the free pro duct of a ﬁnite num b er of copies of Z ) into G . T o simplify the exp osition, we shall use the same nota- tion Hol to denote its quotient map induced b y the canonical pro jection C 1 ( Γ ; G ) → C 1 ( Γ ; G ) /C 0 ( Γ ; G ). Theorem 2.1 b elo w establishes the bijectivit y of the quotien t map. Theorem 2.1 If Γ is c onne cte d, the map Hol : C 1 ( Γ ; G ) /C 0 ( Γ ; G ) → Hom ( π 1 ( Γ ) , G ) /G deﬁne d as Hol ([ ρ ]) = [hol ρ ] is bije ctive. Mor e over, Hom ( π 1 ( Γ ) , G ) /G is in one-to-one c orr esp ondenc e with e quivalenc e classes of ﬂat princip al G -bund les B ρ with ρ ∈ C 1 ( Γ ; G ) . The Geometry of Synchronization Problems and Learning Group Actions 13 Pr o of W e construct an in v erse of Hol from Hom ( π 1 ( Γ ) , G ) /G back to C 1 ( Γ ; G ) /C 0 ( Γ ; G ). Fix an arbitrary v ertex v 0 ∈ Γ , and let χ : π 1 ( Γ , v 0 ) → G b e a group homomorphism. By the connectivity of Γ , eac h v ertex v i ∈ V of Γ is connected to v 0 through an oriented path γ 0 i ; we orient these paths so they all start at vertex v 0 , and enforce γ 00 = { v 0 } . Assign to eac h edge ( i, j ) ∈ E an element ρ ij of the group G deﬁned b y ρ ij := χ  γ 0 i ◦ ( i, j ) ◦ γ − 1 0 j  . (18) Clearly , ρ ij = ρ − 1 j i follo ws from the fact that χ is a group homomorphism; so do es ρ ii = e for all vertices v i ∈ V . Of course, an edge p otential ρ deﬁned as in (18) depends on the choice of the orien ted paths { γ 0 i } ; this dependence is remov ed after passing to the orbit space [ ρ ] ∈ C 1 ( Γ ; G ) /C 0 ( Γ ; G ). In fact, let { ˜ γ 0 i } be an arbitrary choice of | V | orien ted paths connecting v 0 to eac h vertex of Γ satisfying ˜ γ 00 = { v 0 } , then ˜ ρ ij = χ  ˜ γ 0 i ◦ ( i, j ) ◦ ˜ γ − 1 0 j  = χ  ˜ γ 0 i ◦ γ − 1 0 i  χ  γ 0 i ◦ ( i, j ) ◦ γ − 1 0 j  χ  γ 0 j ◦ ˜ γ − 1 0 j  = χ  γ 0 i ◦ ˜ γ − 1 0 i  ρ ij χ  γ 0 j ◦ ˜ γ − 1 0 j  , i.e., as elements in C 1 ( Γ ; G ), ˜ ρ diﬀers from ρ b y an action of the v ertex potential f ∈ C 0 ( Γ ; G ) deﬁned as f i := χ  γ 0 i ◦ ˜ γ − 1 0 i  , ∀ v i ∈ V . Therefore, (18) uniquely speciﬁes an element [ ρ ] in C 1 ( Γ ; G ) /C 0 ( Γ ; G ) for any χ ∈ Hom ( π 1 ( Γ ) , G ). It remains to show that Hol ([ ρ ]) diﬀers from χ b y an inner automorphism of G . T o see this, let ω b e an arbitrary oriented lo op on Γ based at v 0 consisting of consecutive edges ( v 0 , v i 1 ) , ( v i 1 , v i 2 ) , · · · , ( v i N , v 0 ), where N is some nonnegative integer. Using γ 00 = { 0 } and γ − 1 0 i ◦ γ 0 i = { v 0 } for any v i ∈ V , w e ha v e hol ρ ( ω ) =  ρ 0 ,i N ρ i N ,i N − 1 · · · ρ i 2 ,i 1 ρ i 1 , 0  − 1 = ρ 0 ,i 1 ρ i 1 ,i 2 · · · ρ i N − 1 ,i N ρ i N , 0 = χ  γ 00 ◦ ( v 0 , v i 1 ) ◦ γ − 1 0 ,i 1  χ  γ 0 ,i 1 ◦ ( v i 1 , v i 2 ) ◦ γ − 1 0 ,i 2  · · · χ  γ 0 ,i N − 1 ◦  v i N − 1 , v i N  ◦ γ − 1 0 ,i N  χ  γ 0 ,i N ◦ ( v i N , v 0 ) ◦ γ − 1 00  = χ ( { v 0 } ) χ (( v 0 , v i 1 ) ◦ ( v i 1 , v i 2 ) ◦ · · · ◦ ( v i N , v 0 )) χ ( { v 0 } ) − 1 = χ ( ω ) . This calculation is clearly indep endent of the c hoices of orien ted paths { ˜ γ 0 i } . Th us Hol maps [ ρ ] exactly to χ , an elemen t of Hom ( π 1 ( Γ , v 0 ) , G ); the indep endence of Hol ([ ρ ]) as an element of Hom ( π 1 ( Γ ) , G ) /G with resp ect to the c hoice of the base p oin t v 0 follo ws from an essentially identical argumen t as given in the pro of of Lemma 2.3. The last statemen t follo ws from Lemma 2.1. Theorem 2.1 is closely related to classiﬁcation theorems of ﬂat connections and principal bundles with disconnected structure group (see, e.g. [144, § 13.6]) and [141, § 13.9]). The synchronizabilit y of an edge p oten tial ρ on connected graph Γ , whic h is equiv alent to the triviality of B ρ (c.f. Prop osition 1.2), can no w b e interpreted as the corresp onding conjugacy class of Hom ( π 1 ( Γ ) , G ). In fact, the conjugacy class corresp onding to trivial bundles B ρ is also trivial and reﬂects the trivialit y of the holonomy of Γ . The pro of of Corollary 2.1 further develops this observ ation. Corollary 2.1 F or a c onne cte d gr aph Γ and top olo gic al gr oup G , an e dge p otential ρ ∈ C 1 ( Γ ; G ) is syn- chr onizable if and only if Hol ρ ( Γ ) is trivial. Pr o of Note that ρ ∈ C 1 ( Γ ; G ) is sync hronizable (see (1)) if and only if there exists f ∈ C 0 ( Γ ; G ) suc h that f − 1 i ρ ij f j = e ∈ G ∀ ( i, j ) ∈ E , where e is the identify element of the structure group G . This is equiv alen t to saying that [ ρ ] = [ e ] ∈ C 1 ( Γ ; G ) /C 0 ( Γ ; G ) , where e ∈ C 1 ( Γ ; G ) is deﬁned as e ij = e ∈ G for all ( i, j ) ∈ E , (19) whic h b y Theorem 2.1 implies Hol ([ ρ ]) = Hol ([ e ]) = Id e ∈ Hom ( π 1 ( Γ ) , G ) /G, where Id e : π 1 ( Γ ) → G is the constant map sending all oriented lo ops in Γ to the iden tity elemen t e ∈ G . The conclusion follows immediately b y noting that Hol ([ ρ ]) = Id e ⇔ for an y v ∈ V , hol ρ ( ω ) = e for all orien ted loops ω based at v ⇔ Hol ρ ( v ) = { e } for all vertices v ∈ V ⇔ Hol ρ ( Γ ) is trivial where for the last equiv alence we inv oked Lemma 2.4. 14 Tingran Gao, Jacek Bro dzki, and Say an Mukherjee Corollary 2.1 on its own can b e deriv ed from an elemen tary argumen t. In fact, without descending hol ρ from Ω v to π 1 ( Γ , v ), we can still deﬁne Hol ρ ( v ) as the image hol ρ ( Ω v ), though hol ρ is not injectiv e as a group homomorphism from Ω v to G . The triviality of Hol ρ ( v ) still implies the existence of a vertex p oten tial f ∈ C 0 ( Γ ; G ) compatible with ρ ∈ C 1 ( Γ ; G ) (simply b y setting f i = e on an arbitrarily chosen v i ∈ V and progressiv ely propagating v alues of f to neighboring v ertices), and vic e versa . The exp osition in this section, cen tered around Theorem 2.1, extends b eyond this elementary argument and strives to unv eil a complete geometric picture underlying the “corresp ondence betw een trivialities” discussed in Corollary 2.1. In future w ork w e intend to pursue nov el synchronization algorithms based on metric and symplectic structures on the moduli space of ﬂat bundles (see, e.g. [7, 155, 80]). 2.2 A Twisted Hodge Theory for Sync hronization Problems In this section w e relate sync hronization to the ﬁrst cohomology of a de Rham cochain complex on Γ with co eﬃcien ts twisted by a representation space F of the structure group G . This can b e in terpreted as an instance of the standard de Rham cohomology of ﬂat bundles (see e.g. [157, 70, 132, 133]). The ﬁbre bundles considered in this section are vector bundles (with ﬁbre type F ) asso ciated with the principal bundle studied in Section 2.1. When the v ector space F is equipp ed with a metric, the vector bundle inherits a compatible metric, with whic h a t wisted Ho dge Laplacian can b e constructed; sp ecial cases of this twisted Laplacian in the low est degree include the connection Laplacian [136, 13]. In this setting, synchronizabilit y is realized as a condition on the dimension of the null space of the lo west degree t wisted Ho dge Laplacian, t his is reminiscen t of the classical Riemann-Hilb ert corresp ondence betw een ﬂat connections and lo cally constan t sheav es. The sp ectral information of the twisted Ho dge Laplacian serv es as a quantiﬁcation of the the level of obstruction to sync hronizabilit y . 2.2.1 Flat Asso ciate d Bund les and Twiste d Zer o-F orms Let B ρ b e the sync hronization principal bundle on Γ asso ciated with ρ ∈ C 1 ( Γ ; G ), as in Prop osition 1.2, and F b e a topological space on whic h G acts on the left as a topological transformation group. Denote the action of G on F as τ : G → Aut ( F ). Consider the righ t action of G on B ρ × F as ( p, v ) 7− →  pg , τ  g − 1  v  . The orbit space of this action, con v entionally denoted as B ρ × G F or B ρ [ F ], is referred to as the F -bund le asso ciate d with princip al bund le B ρ , or asso ciate d F -bund le for short. W e will denote the bundle pro jection as π : B ρ [ F ] → Γ , and denote B ρ [ F ] x := π − 1 ( x ) for the ﬁbr e over x ∈ Γ . Strictly sp eaking, the graph Γ should b e distinguished from its underlying top ological space, but w e use the same notation Γ for b oth as long as the meaning is clear from the context. The same op en cov er U of Γ that trivializes B ρ also trivializes B ρ [ F ]. In fact, the bundle transition function of B ρ [ F ] on an y nonempty U i ∩ U j is the constan t map U i ∩ U j → τ ( ρ ij ) ∈ Aut ( F ), where U i ∩ U j → ρ ij is the constant bundle transition function from U i to U j for B ρ . Consequently , the associated bundle B ρ [ F ] is also ﬂat. Unless confusions arise, w e shall refer to B ρ [ F ] as the ﬂat asso ciate d F -bund le of B ρ , and denote the lo cal trivialization of the asso ciated bundle ov er U i ∈ U using the same notation φ i : U i × F → B ρ [ F ] as for the principal bundle B ρ . In the con text of synchronization problems, the most relev an t asso ciated bundles are those with ﬁbre F b eing a v ector space and structure group G b eing the general linear group GL ( F ). These types of ﬁbre bundles are commonly referred to as ve ctor bund les . W e will focus on ﬂat associated vector bundles for the rest of the section, though the deﬁnition of ﬁbr e pr oje ctions and se ctions extend literally to general ﬁbre bundles. F or simplicity of presen tation, we will omit the notation τ and write the bundle transition functions again as ρ ij (instead of τ ( ρ ij )), since its action on a vector space F is simply matrix-vector multi plication. W e no w focus on se ctions , the analog of “functions” on smooth manifolds but with v alues in ﬁbre bundles. F or a general ﬁbre bundle E → B , a lo c al se ction s : U → E | U of E on an op en set U of the base space B is a contin uous map from U to E | U suc h that π ◦ s is identiﬁed on U . A glob al se ction of E is a lo cal section deﬁned on the entire base space B . W e shall enco de the data of synchronization problems in to the language of sections of ﬂat asso ciated bundles. The discrete nature of the problem naturally motiv ates us to The Geometry of Synchronization Problems and Learning Group Actions 15 consider sp ecial classes of local and global sections that are “constant” within each op en set in U , in a sense to b e made clear so on in local coordinates. The follo wing notion of ﬁbr e pr oje ction is introduced to simplify notations in v olving local coordinates. Deﬁnition 2.1 F or an y i ∈ V , deﬁne the ﬁbr e pr oje ction ov er U i ∈ U , denoted as p i : B ρ [ F ]   U i = π − 1 ( U i ) − → F , (20) as the comp osition of φ − 1 i : B ρ [ F ]   U i → U i × F with the canonical pro jection U i × F → F . F or any x ∈ U i , the restriction of p i to the ﬁbre B ρ [ F ] x , denoted as p i,x : B ρ [ F ] x → F , is (b y deﬁnition) sim ultaneously a homeomorphism betw een top ological spaces and an isomorphism b etw een vector spaces. Deﬁnition 2.2 (Constan t Lo cal Sections) A c onstant lo c al se ction s : U i → B ρ [ F ]   U i of the bundle B ρ [ F ] on open set U i ∈ U is a local section of B ρ [ F ] such that p i,x ( s ( x )) is a constant elemen t of F for all x ∈ U i . W e refer to the linear space of all constan t local sections on U i ∈ U as c onstant twiste d lo c al 0 -forms on U i , denoted as Ω 0 i ( Γ ; B ρ [ F ]). Clearly , a constant local section s ∈ Ω 0 i ( Γ ; B ρ [ F ]) is unambiguously determined b y ev aluating s at v ertex i , or equiv alen tly by reading oﬀ the ﬁbre pro jection image s i := p i ( s ( i )). W e denote this characterization of s as p i ( s ( x )) ≡ s i , ∀ x ∈ U i , s ∈ Ω 0 i ( Γ ; B ρ [ F ]) . (21) When we consider x ∈ U i ∩ U j where U i ∩ U j 6 = ∅ (i.e. when ( i, j ) ∈ E ), it will b e conv enien t to note that the ﬁbre pro jection p j ev aluates s ( x ) to p j ( s ( x )) = p j ◦ p − 1 i ( p i ( s ( x ))) ≡ ρ j i s i . This can b e understo o d as a “c hange-of-coordinates” form ula for constant lo cal sections. Let C 0 ( Γ ; F ) := { f : V → F } denote the linear space of F -value d 0 -c o chains on graph Γ . Every element f of C 0 ( Γ ; F ) deﬁnes a collection of constan t lo cal sections  f ( i ) : U i → π − 1 ( U i ) | U i ∈ U  , one for each U i ∈ U with f ( i ) ( x ) := p − 1 i,x ( f i ) , ∀ x ∈ U i . (22) W e th us hav e the canonical iden tiﬁcation C 0 ( Γ ; F ) = Y i ∈ V Ω 0 i ( Γ ; B ρ [ F ]) . (23) Of course, the constant lo cal sections  f ( i )  sp eciﬁed b y f ∈ C 0 ( Γ ; F ) generally do not give rise to a global section of the bundle B ρ [ F ], unless they “patc h together” seamlessly on ev ery nonempt y in tersection U i ∩ U j , satisfying the condition p − 1 i,x ( f i ) = f ( i ) ( x ) = f ( j ) ( x ) = p − 1 j,x ( f j ) , ∀ x ∈ U i ∩ U j ⇔ f i = p i,x ◦ p − 1 j,x ( f j ) = ρ ij f j , ∀ x ∈ U i ∩ U j . (24) The right hand side of (24) is recognized as a solution to the sync hronization problem with prescrib ed edge p oten tial ρ . W e ha ve thus prov ed the follo wing Lemma. Lemma 2.5 The c onstant lo c al se ctions sp e ciﬁe d by f ∈ C 0 ( Γ ; F ) deﬁne a glob al se ction on B ρ [ F ] if and only if f i = ρ ij f j , ∀ ( i, j ) ∈ E , (25) i.e., if and only if the vertex p otential f : V → F is a solution to the F -synchr onization pr oblem over Γ with r esp e ct to the e dge p otential ρ ∈ C 1 ( Γ ; G ) . When condition (25) is satisﬁed, the resulting global section constructed from constant local sections is sp ecial among all global sections of B ρ [ F ] in that its restriction to each U i has constan t image under ﬁbre pro jection ov er U i . This type of global section will b e of ma jor interest in the remainder of this section. Deﬁnition 2.3 (Lo cally Constant Global Section) A global section s : Γ → B ρ [ F ] is said to b e lo c al ly c onstant if p i ( s ( x )) ≡ const . ∀ x ∈ U i . (26) The linear space of all lo cally constant global sections on B ρ [ F ] will b e called lo c al ly c onstant twiste d glob al 0 -forms on Γ , denoted as Ω 0 ( Γ ; B ρ [ F ]). 16 Tingran Gao, Jacek Bro dzki, and Say an Mukherjee Naturally , Ω 0 ( Γ ; B ρ [ F ]) em beds into C 0 ( Γ ; F ) b y Ω 0 ( Γ ; B ρ [ F ])  → Y i ∈ V Ω 0 i ( Γ ; B ρ [ F ]) = C 0 ( Γ ; F ) s 7− → ( s | U 1 , · · · , s | U n ) (27) where n = | V | stands for the total n umber of vertices in Γ . The ob jectiv e of a F -synchronization problem o ver Γ with resp ect to ρ ∈ C 1 ( Γ ; G ) can be interpreted in this geometric framew ork as searching for an elemen t of Ω 0 ( Γ ; B ρ [ F ]) in the feasible domain C 0 ( Γ ; F ). The existence of global sections is crucial information for the structure of a ﬁbre bundle. F or principal bundles B ρ considered in Section 2.1, a single global section dictates the trivialit y of the bundle. Though the triviality of a principal bundle is equiv alent to its asso ciated v ector bundle (see Proposition 2.1), B ρ [ F ] is trivial if and only if it admits d = dim F global sections s 1 , · · · , s d that are line arly indep endent in the sense that s 1 x , · · · , s d x on eac h ﬁbre F x are linearly indep enden t as vectors in F (c.f. [115, Theorem 2.2]). A collection of linearly independent global sections are said to form a glob al fr ame (see e.g. [100, Chapter 5]) for the vector bundle, since they deﬁne a basis (frame) for each ﬁbre. As will be established in Proposition 2.1, the fact that the bundle B ρ [ F ] is ﬂat further reduces its trivialit y to ﬁnding d linearly independent locally constan t global sections, for which linear indep endence only needs to be chec k ed at the vertices of Γ . More precisely , adopting notation s i := p i ( s ( i )) , ∀ i ∈ V for an arbitrary section s of B ρ [ F ], w e deﬁne the linear indep endence of lo cally constant global sections of B ρ [ F ] as follows. Deﬁnition 2.4 A collection of k (1 ≤ k ≤ d = dim F ) lo cally constant global sections s 1 , · · · , s k ∈ Ω 0 ( Γ ; B ρ [ F ]) are said to b e line arly indep endent if s 1 i , · · · , s k i are linearly indep enden t as vectors in F at ev ery vertex i ∈ V . By the em bedding (27), an y s ∈ Ω 0 ( Γ ; B ρ [ F ]) can be equiv alently encoded in to a v ector of dimension nd , where d = dim F and n = | V | stands for the n um b er of vertices of Γ . In fact, just as for an y v ertex p oten tial in C 0 ( Γ ; F ), one simply needs to vertically stack the column vectors { s i = p i s ( i ) | i ∈ V } , the ﬁbre pro jection images of s at each vertex. W e shall write [ s ] for suc h a v ector of length nd that enco des s ∈ Ω 0 ( Γ ; B ρ [ F ]), and refer to the v ector as the r epr esentative ve ctor of the section. The linear independence of lo cally constan t global sections is easily seen to be equiv alent to the linear indep endence of the represen tativ e vectors of length nd , as the follo wing Lemma clariﬁes. Lemma 2.6 A c ol le ction of k ( 1 ≤ k ≤ d = dim F ) lo c al ly c onstant glob al se ctions s 1 , · · · , s k ar e line arly indep endent if and only if  s 1  , · · · ,  s k  ar e line ar indep endent as ve ctors of length nd . Pr o of Since B ρ [ F ] is a ﬂat bundle and the graph Γ is assumed connected, the linear indep endence of lo cally constant global sections s 1 , · · · , s k is equiv alen t to the linear indep endence of vectors s 1 i , · · · , s k i at an y vertex i : since ρ j i ∈ G are all inv ertible, vectors s 1 i , · · · , s k i are linearly indep endent if and only if s 1 j = ρ j i s 1 i , · · · , s k j = ρ j i s k i are linearly indep endent. F or deﬁniteness, let us ﬁx i = 1. W rite S for the nd -by- k matrix with  s j  as its j th column, S 1 for the d × k matrix with s j 1 as its j th column, e = [1 , · · · , 1] > for the column v ector of length d with all entries equal to one, and P for the nd -b y- nd blo c k diagonal matrix with ρ j 1 at its j th diagonal blo c k (adopting the conv en tion ρ 11 = I n × n ). The conclusion follo ws from the matrix iden tity S = P ( e ⊗ S 1 ) and rank ( S ) = rank  [1 , · · · , 1] >  · rank ( S 1 ) = rank ( S 1 ) . R emark 2.2 Note that the equiv alence of tw o notions of linear indep endence only holds if w e already kno w that s 1 , · · · , s k are global sections. F or general f 1 , · · · , f k ∈ C 0 ( Γ ; F ) that are linearly indep endent as nd - v ectors, their corresp onding represen tatives in Q i ∈ V Ω 0 i ( Γ ; B ρ [ F ]) do not necessarily deﬁne global sections, nor are they in general linearly indep endent as constant lo cal sections on each U i . A simple example is to consider a graph Γ consisting of t wo vertices V = { v 1 , v 2 } and only one edge connecting them, F = R 2 , and G is the trivial group consisting of only the 2 × 2 identit y matrix: vectors  f 1  = (1 , 0 , 1 , 0) > and  f 2  = (1 , 0 , 0 , 1) > are linearly indep endent as vectors in R 4 but do not deﬁne linearly indep endent constant lo cal sections on U 1 . The Geometry of Synchronization Problems and Learning Group Actions 17 With all essential concepts presented, we are ready to establish our main observ ation in this subsection. Prop osition 2.1 L et G b e a top olo gic al tr ansformation gr oup acting on a (r e al or c omplex) d -dimensional ve ctor sp ac e F on the left, Γ = ( V , E ) b e a c onne cte d undir e cte d gr aph, and ρ ∈ C 1 ( Γ ; G ) a G -value d e dge p otential. The fol lowing statements ar e e quivalent: (i) B ρ is trivial; (ii) B ρ admits a glob al se ction; (iii) B ρ [ F ] is trivial; (iv) B ρ [ F ] admits d = dim F line arly indep endent lo c al ly c onstant glob al se ctions. Pr o of The equiv alence of (i) and (iii) follows from [141, Theorem 4.3]. The equiv alence (i) ⇔ (ii) follows from standard diﬀerential geometry , see e.g. [141, 115]; similarly standard is the equiv alence b etw een (iii) and the existence of n linearly indep endent global sections on B ρ [ F ]. T o show the equiv alence (iv) ⇔ (iii), it suﬃces to pro ve that a trivial ﬂat v ector bundle B ρ [ F ] admits d linearly independent global sections that are also lo cally constant. T o see this, recall from Prop osition 1.2 and Corollary 2.1 that B ρ [ F ] is trivial ⇔ B ρ is trivial ⇔ Hol ([ ρ ]) is trivial ⇔ ∃ g ∈ Ω 0 ρ ( Γ ; G ) s.t. g − 1 i ρ ij g j = e ∀ ( i, j ) ∈ E ⇔ ∃ g ∈ Ω 0 ρ ( Γ ; G ) s.t. ρ ij = g i g − 1 j ∀ ( i, j ) ∈ E . Let { e 1 , · · · , e d } be a basis for F , and for each k = 1 , · · · , d deﬁne s k : Γ → B ρ [ F ] as s k ( x ) = p − 1 i,x ( g i e k ) ∀ x ∈ U i . It is straigh tforward to verify by deﬁnition that s k is a w ell-deﬁned global section and is lo cally constant. That s 1 , · · · , s d are linearly independent as global sections follows from the fact that p i,x : F → B ρ [ F ] x are isomorphisms betw een vector spaces. R emark 2.3 The global section in (ii) is also “locally constan t” in a sense analogous to Deﬁnition 2.3 but for principal bundles; w e do not in tro duce this deﬁnition here since global sections on principal bundles will not be pursued directly in this w ork. Prop osition 2.1 points out an alternative approac h for determining the synchronizabilit y of an edge p oten tial ρ ∈ C 1 ( Γ ; G ), at least when G is a matrix group GL ( F ): it suﬃces to chec k the existence of d = dim F linearly indep endent lo cally constan t global sections on the ﬂat asso ciated vector bundle B ρ [ F ]. Suc h existence can be stated as a cohomological obstruction. W e will pursue such a formulation in the next section. In Section 2.2.3 we will utilize the inner pro duct structure on F to reduce the structure group of a GL ( F )-bundle to O ( d ) or U ( d ), as commonly seen in synchronization problems [12, 13, 34]. If the underlying ﬁbre bundle is orien table, the same pro cedures further reduce the structure group to SO ( d ) or SU ( d ), corresponding to synchronization problems considered in [138, 154]. 2.2.2 Twiste d One-F orms and De Rham Cohomolo gy In a smooth category , sections on a ﬁbre bundle can b e diﬀeren tiated by a c ovariant derivative . The resulting ob ject is a sk ew-symmetric “direction-dep enden t” section on the same bundle, or equiv alen tly a section of a new ﬁbre bundle whic h is the tensor pro duct of the original ﬁbre bundle with the bundle of 1-forms on the base manifold. W e shall generalized this picture to the discrete/combinatorial setting for ﬂat associated bundles B ρ [ F ] that naturally arise in sync hronization problems. Recall from discrete Ho dge theory [49, 101, 53, 85, 86] that discrete 0-forms and 1-forms on a graph Γ are deﬁned as Ω 0 ( Γ ) := { f : V → K } , Ω 1 ( Γ ) := { ω : E → K | ω ij = − ω j i ∀ ( i, j ) ∈ E } , where K = C or R , which we will assume to b e the num b er ﬁeld for the v ector space F . Let us deﬁne a lo cal v ersion of Ω 1 ( Γ ) b y Ω 1 i ( Γ ) := { ω : N i → K | ω j k = − ω kj ∀ ( j, k ) ∈ N i } , where N i := { ( j, k ) ∈ E | j = i or k = i } . (28) 18 Tingran Gao, Jacek Bro dzki, and Say an Mukherjee In other words, elements of Ω 1 i ( Γ ) are restrictions of elemen ts of Ω 1 ( Γ ) to U i . By a partition of unit y argumen t, it is straigh tforward to iden tify Ω 1 ( Γ ) with (  ω (1) , · · · , ω ( n )  ∈ Y i ∈ V Ω 1 i ( Γ )    ω ( i ) ij = ω ( j ) ij (= − ω ( j ) j i = − ω ( i ) j i ) ) . (29) Deﬁnition 2.5 (Constan t Lo cal 1 -forms) A c onstant twiste d lo c al 1 -form on op en set U i ∈ U is a lo cal section of Ω 0 i ( Γ ; B ρ [ F ]) ⊗ Ω 1 i ( Γ ). Equiv alently , a constant t wisted lo cal 1-form on U i is a map ω : U i × N i → B ρ [ F ] suc h that: (i) F or an y ( j, k ) ∈ N i , ω j k : U i → B ρ [ F ] is a constan t local section on U i , i.e. p i,x ( ω ij ( x )) ≡ const.; (ii) F or an y x ∈ U i and an y U j ∈ U such that U i ∩ U j 6 = ∅ , ω ij ( x ) = − ω j i ( x ). W e denote the linear space of all constan t t wisted local 1-forms on U i as Ω 1 i ( Γ ; B ρ [ F ]). A similar notion of globally deﬁned twisted 1-forms will also b e of interest. In the discrete setting of sync hronization problems, it suﬃces to consider t wisted global 1-forms that are locally constan t under ﬁbre pro jections. Deﬁnition 2.6 (Lo cally Constan t Global 1 -forms) A lo c al ly c onstant twiste d glob al 1 -form is a section of the tensor pro duct bundle Ω 0 ( Γ ; B ρ [ F ]) ⊗ Ω 1 ( Γ ). In other w ords, a locally constan t t wisted global 1-form is a map ω : { ( i, ( j, k )) | i ∈ V , ( j, k ) ∈ N i } → B ρ [ F ] suc h that: (i) F or an y U i ∈ U , ω | U i is a constant twisted lo cal 1-form on U i ; (ii) F or an y x ∈ U i ∩ U j 6 = ∅ , p i,x ( ω | U i ( x )) = ρ ij p j,x  ω | U j ( x )  . W e denote the linear space of all lo cally constant twisted global 1-forms on Γ as Ω 1 ( Γ ; B ρ [ F ]). The deﬁnition of Ω 1 ( Γ ; B ρ [ F ]) already characterized the condition under which a given collection of constan t twisted lo cal 1-forms  ω ( i ) ∈ Ω 1 i ( Γ ; B ρ [ F ])  , one for each U i ∈ U , can b e patc hed to form a lo cally constan t twisted global 1-form. Similar to (24), it suﬃces to chec k the compatibility under “c hange of coordinates”. Lemma 2.7 A c ol le ction of c onstant twiste d lo c al 1 -forms  ω ( i ) ∈ Ω 1 i ( Γ ; B ρ [ F ])  deﬁnes a lo c al ly c onstant twiste d glob al 1 -form if and only if p i,x  ω ( i ) ij ( x )  = ρ ij p j,x  ω ( j ) ij ( x )  , ∀ ( i, j ) ∈ E . (30) Sinc e b oth sides of e quality (30) ar e c onstants, we shal l simplify (30) as p i  ω ( i ) ij  = ρ ij p j  ω ( j ) ij  , ∀ ( i, j ) ∈ E . (31) A signiﬁcan t diﬀerence b et ween Ω 1 ( Γ ; B ρ [ F ]) and Ω 0 ( Γ ; B ρ [ F ]) is that a locally constan t t wisted global 1-form do es not naturally arise from an F -v alued 1-co c hain in C 1 ( Γ ; F ) := { ω : E → F | ω ij = − ω j i } , and these co c hains play an esse n tial role in the discrete Ho dge theory . F or instance, if we set ω ( i ) ij ( x ) := p − 1 i,x ( ω ij ) =: − ω ( i ) j i ( x ) for all x ∈ U i , then  ω ( i ) | i ∈ V  giv es rise to a t wisted global 1-form if and only if ω ij = p i,x  ω ( i ) ij ( x )  = ρ ij p j,x  ω ( j ) ij ( x )  = − ρ ij p j,x  ω ( j ) j i ( x )  = − ρ ij ω j i , ∀ ( i, j ) ∈ E , a condition that is generally not satisﬁed unless ρ ij ≡ e ∈ G for all ( i, j ) ∈ E . This observ ation indicates that C 1 ( Γ ; F ) is not a geometric ob ject naturally asso ciated with the structure of the vector bundle B ρ [ F ], but rather a sp ecial case of Ω 1 ( Γ ; B ρ [ F ]) when the vector bundle B ρ [ F ] is trivial. In this case ρ ij = g i g − 1 j , ∀ ( i, j ) ∈ E for a G -v alued vertex p otential g : V → G and the “gauge-transformed” constant twisted The Geometry of Synchronization Problems and Learning Group Actions 19 lo cal 1-forms  p − 1 i,x ( g i ω ij ) | i ∈ V , ( i, j ) ∈ E  satisfy the compatibility condition (30). Exploring the action of the gauge group on t wisted forms will be considered in the future. With an appropriate notion of twisted 1-forms, we are ready to deﬁne the t wisted diﬀeren tial op erator on twisted 0-forms. This operation is a discrete analog of the cov ariant deriv ativ es for smo oth ﬁbre bundles, and in the mean while, a ﬁbre bundle analog of the discrete exterior deriv ativ e in discrete Hodge theory . Deﬁnition 2.7 (Twisted Diﬀeren tial on Twisted 0 -Co c hains) F or ρ ∈ C 1 ( Γ ; G ) and U i ∈ U , the ρ -twiste d diﬀer ential is a linear operator taking an y f ∈ C 0 ( Γ ; F ) = Q i ∈ V Ω 0 i ( Γ ; B ρ [ F ]) to a collection of n constan t twisted lo cal 1-forms, one for eac h U i ∈ U d ρ : Y i ∈ V Ω 0 i ( Γ ; B ρ [ F ]) − → Y i ∈ V Ω 1 i ( Γ ; B ρ [ F ]) f 7− →  ( d ρ f ) (1) , · · · , ( d ρ f ) n  where eac h ( d ρ f ) ( i ) ∈ Ω 1 i ( Γ ; B ρ [ F ]) is deﬁned as ( d ρ f ) ( i ) ij ( x ) := p − 1 i,x ( f i − ρ ij f j ) =: − ( d ρ f ) ( i ) j i ( x ) , ∀ U i ∈ U , ∀ x ∈ U i ∩ U j 6 = ∅ , f ∈ Ω 0 ρ ( Γ ; F ) . (32) Though d ρ is deﬁned as a linear operator mapping into a collection of constan t t wisted lo cal 1-forms, a somewhat surprising fact is that these constant twisted lo cal 1-forms do patch together to form an elemen t of Ω 1 ( Γ ; B ρ [ F ]). Prop osition 2.2 The twiste d diﬀer ential d ρ maps C 0 ( Γ ; F ) into Ω 1 ( Γ ; B ρ [ F ]) . Pr o of It suﬃces to c heck (31) for the collection of constant t wisted local 1-forms n ( d ρ f ) ( i )   i = 1 , · · · , n o . In fact, p i  ( d ρ f ) ( i ) ij  = f i − ρ ij f j = − ρ ij ( f j − ρ j i f i ) = − ρ ij p j  ( d ρ f ) ( j ) j i  = ρ ij p j  ( d ρ f ) ( j ) ij  ∀ ( i, j ) ∈ E . Since the graph Γ (viewed as a simplicial complex) do es not con tain any 2-simplices, d ρ is the only diﬀeren tial needed for sp ecifying the ρ -twiste d chain c omplex 0 − → C 0 ( Γ ; F ) d ρ − → Ω 1 ( Γ ; B ρ [ F ]) − → 0 . (33) The only non-trivial cohomology group in this de Rham-type chain complex is at the 0-th order H 0 ρ ( Γ ; B ρ [ F ]) := ker d ρ . Prop osition 2.3 k er d ρ = Ω 0 ( Γ ; B ρ [ F ]) . Pr o of Note in the deﬁnition (32) that f ∈ k er d ρ ⇔ f i = ρ ij f j , ∀ ( i, j ) ∈ E . The conclusion then follo ws from Lemma 2.5. By Prop osition 2.1, detecting the synchronizabilit y of ρ ∈ C 1 ( Γ ; G ) no w reduces to chec king if dim ker d ρ = dim F holds. F urthermore, in scenarios where this dimension equalit y do es not hold, dim ker d ρ still provides a quantitativ e measure for the extent to which sync hronizabilit y fails. In this sense, the cohomology group H 0 ρ ( Γ ; F ) serv es as the opposite of a “top ological obstruction” to the synchronizabilit y of ρ ∈ C 1 ( Γ ; G ). 20 Tingran Gao, Jacek Bro dzki, and Say an Mukherjee 2.2.3 Twiste d Ho dge The ory and Synchr onizability In the remainder of this section, w e will fo cus on ﬂat asso ciated bundles B ρ [ F ] with the vector space F equipp ed with an inner product h· , ·i F : F × F → K , where K = C or R dep ending on the v ector space F . This inner pro duct on F will be further assumed with G -invarianc e , in the sense that h g x, g y i F = h x, y i F ∀ x, y ∈ F , g ∈ G. In the terminology of representation theory , we assume that the representation of G on F is unitary (c.f. [25, § II.1]). This inner product in tro duces other related concepts into the geometric framew ork: – F is equipped with a G -inv ariant norm k x k F = h x, x i F for all x ∈ F , whic h further induces an operator norm on G via duality k g k := sup m ∈ F k m k6 =0 k g m k k m k , ∀ g ∈ G. F or simplicit y , we will use the same notation for the norm on F and the dual norm on G . – F or an y g ∈ G , its formal adjoint with resp ect to the inner product h· , ·i F , denoted as g ∗ , is deﬁned as h g x, y i = h x, g ∗ y i ∀ x, y ∈ F . Note that k g ∗ k = k g k for any g ∈ G . – The t wisted 0-cochains C 0 ( Γ ; F ) and locally constant twisted global 1-forms Ω 1 ( Γ ; B ρ [ F ]) are equipped with inner pro ducts and norms induced from the G -inv ariant inner pro duct h· , ·i F , as follows: h f , g i := X i ∈ V d i h f i , g i i F , ∀ f , g ∈ C 0 ( Γ ; F ) (34) h ω , η i := 1 2 X i ∈ V X j :( i,j ) ∈ E w ij D p i  ω ( i ) ij  , p i  η ( i ) ij E F , ∀ ω , η ∈ Ω 1 ( Γ ; B ρ [ F ]) (35) k f k := h f , f i 1 2 , k ω k := h ω , ω i 1 2 , ∀ f ∈ C 0 ( Γ ; F ) , ω ∈ Ω 1 ( Γ ; B ρ [ F ]) (36) where w ij is the weigh t on edge ( i, j ) ∈ E and d i = P j :( i,j ) ∈ E w ij is the weigh ted degree of vertex v i . Note that by the G -inv ariance of h· , ·i F the sum in (35) can b e equiv alently written as (see Appendix A for a quick calculation) h ω , η i = X ( i,j ) ∈ E w ij D p i  ω ( i ) ij  , p i  η ( i ) ij E F . (37) Through lo cal trivializations, an inner pro duct on the vector space F also induces a bund le metric on B ρ [ F ], i.e., a section of the second symmetric p ow er of the dual bundle of B ρ [ F ] which restricts to eac h ﬁbre as a symmetric p ositive deﬁnite quadratic form. As is well known (see e.g. [144, Chapter 7]), a bundle metric can b e used to reduce the structure group of a v ector bundle from GL ( F ) to O ( d ) or U ( d ), where d = dim ( F ). It suﬃces to consider global sections of B ρ [ F ] for ρ ∈ C 1 ( Γ ; O ( d )) or ρ ∈ C 1 ( Γ ; U ( d )) for man y synchronization problems of practical interest [13, 12, 34], instead of requiring ρ ∈ C 1 ( Γ ; GL ( d ; R )) or ρ ∈ C 1 ( Γ ; GL ( d ; C )). Other imp ortan t types of sync hronization problem in volving SO ( d ) and SU ( d ) can b e treated in this geometric framework as determining global sections of orientable v ector bundles (see e.g. [144, Chapter 7] or [20, Proposition 6.4]). Also note that ρ − 1 ij = ρ ∗ ij for edge p otentials in all these sp ecial matrix groups. Since the bundle reduction allo ws us to fo cus only on sync hronization problems with orthogonal or unitary matrices, without loss of generalit y , we will alw ays assume the edge potentials satisfy ρ j i = ρ − 1 ij = ρ ∗ ij for all ( i, j ) ∈ E . The same is assumed in [13, 12]. The inner pro duct structures on C 0 ( Γ ; F ) and Ω 1 ( Γ ; B ρ [ F ]) enable us to deﬁne the ρ -twiste d c o diﬀer en- tial δ ρ : Ω 1 ( Γ ; B ρ [ F ]) → C 0 ( Γ ; F ), the formal adjoin t op erator of the twisted diﬀerential d ρ : C 0 ( Γ ; F ) → Ω 1 ( Γ ; B ρ [ F ]) in the chain complex (33), even tually leading to a t wisted Ho dge theory for synchronization problems. The deﬁnition of δ ρ is consistent with the discrete divergence op erator in discrete Ho dge theory [85, 86]: δ ρ : Ω 1 ( Γ ; B ρ [ F ]) − → C 0 ( Γ ; F ) θ 7− →  ( δ ρ θ )   U 1 , · · · , ( δ ρ θ )   U n  (38) The Geometry of Synchronization Problems and Learning Group Actions 21 where eac h ( δ ρ θ )   U i ∈ is deﬁned b y ( δ ρ θ )   U i ( x ) = p − 1 i,x   1 d i X j :( i,j ) ∈ E w ij p i  θ ( i ) ij    ∀ x ∈ U i , θ ∈ Ω 1 ( Γ ; B ρ [ F ]) (39) or equiv alently ( δ ρ θ ) i = p i  ( δ ρ θ )   U i ( i )  = 1 d i X j :( i,j ) ∈ E w ij p i  θ ( i ) ij  ∀ i ∈ V , θ ∈ Ω 1 ( Γ ; B ρ [ F ]) . (40) Prop osition 2.4 With r esp e ct to the inner pr o ducts (34) and (35) , the twiste d c o diﬀer ential δ ρ : Ω 1 ( Γ ; B ρ [ F ]) → C 0 ( Γ ; F ) deﬁne d by (39) is the formal adjoint of the twiste d diﬀer ential d ρ : C 0 ( Γ ; F ) → Ω 1 ( Γ ; B ρ [ F ]) deﬁne d by (32) . Pr o of Note that for any f ∈ C 0 ( Γ ; F ), θ ∈ Ω 1 ( Γ ; B ρ [ F ]), h f , δ ρ θ i = X i ∈ V d i * f i , 1 d i X j :( i,j ) ∈ E w ij p i  θ ( i ) ij  + F = X i ∈ V X j :( i,j ) ∈ E D f i , w ij p i  θ ( i ) ij E F = X ( i,j ) ∈ E hD f i , w ij p i  θ ( i ) ij E F + D f j , w j i p j  θ ( j ) j i E F i = X ( i,j ) ∈ E D f i , w ij p i  θ ( i ) ij E F + X ( i,j ) ∈ E D f j , w j i p j  θ ( j ) j i E F =: ( I ) + ( I I ) . W e k eep the term ( I ) intact and manipulate term ( I I ) using w ij = w j i and the G -inv ariance of h· , ·i F : ( I I ) = X ( i,j ) ∈ E D ρ ij f j , w j i ρ ij p j  θ ( j ) j i E F ( ∗ ) = = X ( i,j ) ∈ E D ρ ij f j , w ij p i  θ ( i ) j i E F ( ∗∗ ) = = − X ( i,j ) ∈ E D ρ ij f j , w ij p i  θ ( i ) ij E F , where we used ρ ij p j  θ ( j ) j i  = p i  θ ( i ) j i  (the compatibility condition (31)) at ( ∗ ), and the skew-symmetry θ ( i ) j i = − θ ( i ) ij at ( ∗∗ ). Re-combining ( I ) and ( I I ), w e conclude that h f , δ ρ θ i = ( I ) + ( I I ) = X ( i,j ) ∈ E D f i , w ij p i  θ ( i ) ij E F − X ( i,j ) ∈ E D ρ ij f j , w ij p i  θ ( i ) ij E F = X ( i,j ) ∈ E D f i − ρ ij f j , w ij p i  θ ( i ) ij E F = X ( i,j ) ∈ E w ij D p i  ( d ρ f ) ( i ) ij  , p i  θ ( i ) ij E F = h d ρ f , θ i . The c hain complex (33) is now also equipped with formal adjoin ts: 0 − − → ← − − C 0 ( Γ ; F ) d ρ − − → ← − − δ ρ Ω 1 ( Γ ; B ρ [ F ]) − − → ← − − 0 . (41) Tw o twiste d Ho dge L aplacians can b e constructed from this chain complex: ∆ (0) ρ := δ ρ d ρ : C 0 ( Γ ; F ) − → C 0 ( Γ ; F ) , (42) ∆ (1) ρ := d ρ δ ρ : Ω 1 ( Γ ; B ρ [ F ]) − → Ω 1 ( Γ ; B ρ [ F ]) . (43) It is straigh tforw ard to see from these deﬁnitions that both t wisted Laplacians are p ositiv e deﬁnite. In view of Hodge theory , it w ould be of in terest to in vestigate the harmonic forms in the complex (41), the kernels of ∆ (0) ρ and ∆ (1) ρ . Lemma 2.8 k er d ρ = k er ∆ (0) ρ and ker δ ρ = k er ∆ (1) ρ . 22 Tingran Gao, Jacek Bro dzki, and Say an Mukherjee Pr o of Clearly k er d ρ ⊂ k er ∆ (0) ρ . F or the rev erse inclusion, note that by adjointness 0 = D f , ∆ (0) ρ f E = k d ρ f k 2 ∀ f ∈ k er ∆ (0) ρ , whic h implies d ρ f = 0. The equality inv olving ker ∆ (1) ρ follo ws from a similar argumen t. The follo wing decomp osition results follow from standard Ho dge-theoretic argumen ts. Theorem 2.2 C 0 ( Γ ; F ) = ker ∆ (0) ρ ⊕ im δ ρ = k er d ρ ⊕ im δ ρ , Ω 1 ( Γ ; B ρ [ F ]) = im d ρ ⊕ k er ∆ (1) ρ = im d ρ ⊕ k er δ ρ . Pr o of W e only present the pro of for the decomp osition of C 0 ( Γ ; F ); the decomp osition for Ω 1 ( Γ ; F ) is similar. First note that b oth C 0 ( Γ ; F ) and Ω 1 ( Γ ; B ρ [ F ]) are ﬁnite dimensional. The subspace ker d ρ and im δ ρ are orthogonal with respect to the inner product (34), since if f ∈ ker d ρ and δ ρ θ ∈ im δ ρ , h f , δ ρ θ i = h d ρ f , θ i = 0 . It remains to prov e that each f ∈ C 0 ( Γ ; F ) can b e decomp osed in to a linear combination of elements in ker ∆ (0) ρ and im δ δ . If d ρ f = 0, the decomp osition is trivial. Otherwise, consider the following P oisson equation: ∆ (1) ρ θ = d ρ f . (44) W e claim that equation (44) has a solution θ ∈ Ω 1 ( Γ ; B ρ [ F ]) as long as d ρ f 6 = 0. In fact, by F redholm alternativ e (see e.g. an exposition for the ﬁnite dimensional case in [101] which suﬃces for our purp ose), if d ρ f / ∈ im ∆ (1) ρ , then d ρ f ∈ k er ∆ (1) ρ = k er δ ρ ; ho w ever, d ρ f ⊥ k er δ ρ since h d ρ f , ω i = h f , δ ρ ω i = 0 ∀ ω ∈ ker δ ρ . This pro v es that (44) has a solution θ ∈ Ω 1 ( Γ ; B ρ [ F ]) for d ρ f 6 = 0. W e can th us split f ∈ C 0 ( Γ ; F ) in to f = ( f − δ ρ θ ) + δ ρ θ , in whic h δ ρ θ ∈ im δ ρ , and f − δ ρ θ ∈ k er d ρ since d ρ ( f − δ ρ θ ) = d ρ f − d ρ δ ρ θ = d ρ f − ∆ (1) ρ θ = 0 . R emark 2.4 Prop osition 2.3 and Theorem 2.2 completely characterized the em b edding (27): the orthogonalit y complemen t (with resp ect to the inner pro duct (34)) of the linear space of solutions to the F -synchronization problem on Γ with respect to ρ ∈ Ω 1 ( Γ ; G ) is exactly the image of the t wisted codiﬀerential (39). In fact, one recognizes from C 0 ( Γ ; F ) = ker ∆ (0) ρ ⊕ im δ ρ , k er ∆ (0) ρ = ker d ρ , and H 0 ρ ( Γ ; B ρ [ F ]) = ker d ρ the w ell-kno wn Ho dge theorem H 0 ρ ( Γ ; B ρ [ F ]) = ker ∆ (0) ρ . 2.2.4 Gr aph Conne ction L aplacian and Che e ger-T yp e Ine qualities for Gr aph F rustr ation In this section, w e connect our geometric framework to the computational aspects of sync hronization algo- rithms. As pointed out in Section 2.2.2, for cases where G = O ( d ) or G = U ( d ), the synchronizabilit y of an edge p otential ρ ∈ C 1 ( Γ ; G ) is equiv alen t to whether or not the equality dim ker d ρ = d holds. Lemma 2.8 reduces the sync hronizability further to the dimension of ker ∆ (0) ρ . With iden tiﬁcation (23), it can b e noticed that ∆ (0) ρ is exactly the gr aph c onne ction L aplacian (GCL) in the literature of synchronization problems, random matrix theory , and manifold learning (see e.g. [136, 13, 97, 58]). Recall from [13] that the graph connection Laplacian for graph Γ and edge p otential ρ ∈ C 1 ( Γ ; G ) is deﬁned as L 1 = D 1 − W 1 (45) where W 1 ∈ K nd × nd is a n × n blo c k matrix with w ij ρ ij ∈ K d × d at its ( i, j )th blo ck, and D 1 ∈ K nd × nd is blo c k diagonal with d i I d × d ∈ K d × d at its ( i, i )th block. T o see that ∆ (0) ρ coincides with L 1 , notice that D f , ∆ (0) ρ f E = k d ρ f k 2 = X ( i,j ) ∈ E w ij k f i − ρ ij f j k 2 = 1 2 X i,j ∈ V w ij k f i − ρ ij f j k 2 = 1 2 [ f ] > L 1 [ f ] , ∀ f ∈ C 0 ( Γ ; F ) . Theorem 2.2 translates into this combinatorial setting as a decomp osition result for the matrix L 1 , as presen ted below in Prop osition 2.5. W e denote n = | V | and m = | E | for the graph Γ = ( V , E ). The Geometry of Synchronization Problems and Learning Group Actions 23 Prop osition 2.5 The gr aph c onne ction L aplacian L 1 ∈ K nd × nd admits a de c omp osition L 1 = [ δ ρ ] [ d ρ ] , [ δ ρ ] ∈ K nd × md , [ d ρ ] ∈ K md × nd , (46) wher e [ d ρ ] is an m -by- n blo ck matrix in which the ( i, j ) th blo ck is given by [ d ρ ] ij =      I d × d if e dge i starts at vertex j , − w kj ρ kj if e dge i starts at vertex k and ends at vertex j , 0 otherwise , (47) and [ δ ρ ] is an n -by- m blo ck matrix in which the ( i, j ) th blo ck is given by [ δ ρ ] ij =    w ik d i I d × d if e dge j starts at vertex i and ends at vertex k , 0 otherwise . (48) Note that her e e ach e dge ( i, j ) app e ars twic e in E with opp osite orientations. The Ho dge decomp osition (46) immediately leads to the following observ ation, which reﬂects the geo- metric fact that there do not exist more than n = dim F linearly indep endent global sections on the v ector bundle B ρ [ F ]. Prop osition 2.6 The dimension of the nul l eigensp ac e of L 1 c an not exc e e d n , the dimension of b oth the c olumn sp ac e of [ d ρ ] and the r ow sp ac e of [ δ ρ ] . By Lemma 2.6, if there are d linearly indep endent v ectors in the k ernel space of L 1 , then they giv e rise to d lo cally constant global sections on B ρ [ F ] that are also linearly indep endent as global sections, which indicates the trivialit y of the vector bundle B ρ [ F ] and the synchronizabilit y of ρ ∈ C 1 ( Γ ; G ). Note that an analogy of this result for graphs with multiple connected comp onen ts also holds, though w e assumed Γ is connected throughout this paper: a graph with k ≥ 1 connected components and a prescrib ed O ( d )-v alued edge potential is sync hronizable if and only if the dimension of the n ull eigenspace of L 1 is kd . This geometric picture is consistent with the main sp ectral relaxation algorithm [13, Algorithm 2.5] when the edge p oten tial is sync hronizable. Basically , the spectral relaxation pro cedure w orks as follows: ﬁrst, extract d eigenv ectors x 1 , · · · , x d corresp onding to the smallest d eigen v alues of L 1 ; second, form the nd × d matrix X =  x 1 , · · · , x d  and split it v ertically into n blo cks X 1 , · · · , X n of equal size d × d ; ﬁnally , ﬁnd the closest orthogonal matrix O i to eac h X i b y p olar decomposition, and construct the desired synchronizing vertex p otential f ∈ C 0 ( Γ ; O ( d )) b y setting f i = O i . Since ker ∆ (0) ρ = d for an y sync hronizable edge potential ρ , the d eigen vectors x 1 , · · · , x d of L 1 all lie in the null eigenspace of L 1 , which provide exactly the d linearly indep enden t global sections needed to trivialize the vector bundle; all that remains for obtaining a desired synchronizing vertex potential is to rescale the columns of each block X i ∈ K d × d to achiev e orthonormalit y , which is exactly what is done in the polar decomp osition step when ρ is synchronizable. The twisted cohomology framew ork dev elop ed in this section suggests the follo wing impro vemen ts when applying the spectral relaxation algorithm for determining sync hronizability of a given edge potential: (1) Instead of c hecking dim ker ∆ (0) ρ , one can simply c heck dim k er d ρ or dim k er δ ρ (whic h gives dim im δ ρ ). Both [ d ρ ] and [ δ ρ ] matrices are muc h smaller in size compared with L 1 , and the dimension can b e determined b y QR decomp osition rather than the more expensive eigen-decomp osition; (2) Instead of p erforming p olar decomp osition for each d × d blo ck X i , which inv olv es the relatively more exp ensiv e SVD, it suﬃces to inv ok e a Gram-Schmidt orthonormalization. If synchronizabilit y of ρ is conﬁrmed b y the dimension test in a previous step, the Gram-Sc hmidt procedure can be p erformed for the entire matrix X ∈ K nd × d in one pass (with a minor mo diﬁcation of k eeping the columns to hav e norm n instead of 1), as opp osed to b eing carried out for eac h individual blo c k X i . R emark 2.5 The Ho dge decomp osition (46) also suggests an alternativ e approach to obtaining n linearly indep enden t locally constan t global sections on B ρ [ F ]: instead of directly solving for the null eigenspace of L 1 , we can lo ok for the orthogonal complement of im [ δ ρ ]. Note, how ev er, that the domain of [ δ ρ ] should not b e taken as the entire K md , since δ ρ is deﬁned on Ω 1 ( Γ ; B ρ [ F ]), in which elements satisfy the compatibility condition (31). Constructing suc h a basis matrix B ∈ K md × md and computing the orthogonal complement of the column space of [ δ ρ ] B turns out not to b e muc h simpler than ﬁnding the orthogonal complement of L 1 (i.e. ﬁnding the n ull eigenspace of L 1 directly). 24 Tingran Gao, Jacek Bro dzki, and Say an Mukherjee In the more general setting where the edge p oten tial ρ is not assumed synchronizable, the geometric picture b ecomes m uch more inv olved. Of central imp ortance to the relaxation algorithms and Cheeger in- equalities in [13] is to minimize the frustr ation of a graph Γ with respect to a prescribed group potential: ν ( Γ ) = inf g ∈ C 0 ( Γ ; O ( d )) ν ( g ) = inf g ∈ C 0 ( Γ ; O ( d )) 1 2 d 1 v ol ( Γ ) X i,j ∈ V w ij k g i − ρ ij g j k 2 F , where v ol ( Γ ) = X i ∈ V d i . (49) As sho wn in (the proof of ) Proposition 2.1, an O ( d )-v alued edge potential ξ ∈ C 1 ( Γ ; O ( d )) is sync hronizable if and only if there exists g ∈ C 0 ( Γ ; O ( d )) such that ξ ij = g i g − 1 j for all ( i, j ) ∈ E . The frustration ν ( Γ ) deﬁned in (49) can thus b e rewritten as ν ( Γ ) = 1 2 d 1 v ol ( Γ ) inf g ∈ C 0 ( Γ ; O ( d )) X i,j ∈ V w ij   g i g − 1 j − ρ ij   2 F = 1 2 d 1 v ol ( Γ ) inf ξ ∈ C 1 sync ( Γ ; O ( d )) X i,j ∈ V w ij k ξ ij − ρ ij k 2 F , where w e deﬁne C 1 sync ( Γ ; O ( d )) :=  ξ ∈ C 1 ( Γ ; O ( d ))   ξ sync hronizable  =  ξ ∈ C 1 ( Γ ; O ( d ))   Hol ξ ( Γ ) is trivial  . Therefore, in the ﬁbre bundle framew ork, a synchronization problem asks for a sync hronizable edge p otential that is “as close as possible” to a prescribe d edge potential, or geometrically speaking, for a trivial ﬂat bundle “as close as p ossible” to a giv en ﬂat bundle. One approach, from the p oin t of view of Prop osition 2.1, is to ﬁnd d linearly indep endent co c hains in C 0  Γ ; R d  that are “as close as p ossible” to b eing a global frame of B ρ  R d  in the sense of minimizing the frustr ation of a S d − 1 -value d c o chain η ( f ) = D f , ∆ (0) ρ f E k f k 2 = 1 2 X i,j ∈ V w ij k f i − ρ ij f j k 2 X i ∈ V d i k f i k 2 = 1 2v ol ( Γ ) [ f ] > L 1 [ f ] , ∀ f ∈ C 0  Γ ; S d − 1  , k f k 6 = 0 , whic h equals zero if and only if f deﬁnes a global section on B ρ  R d  (the constraint k f k 6 = 0 is also in- disp ensable from a geometric p oint of view, as an y v ector bundle trivially admits the constant zero global section). This provides a geometric in terpretation of the sp ectral relaxation algorithm in [13]. F rom a per- turbation p oint of view, the magnitudes of the smallest n eigenv alues of ∆ (0) ρ measure the deviation from degeneracy of the d -dimensional eigenspace of lo w est frequencies, and can thus b e in terpreted as the extent to which B ρ  R d  deviates from admitting d linearly independent global sections and b eing a trivial bun- dle. The Cheeger-type inequality established in [13] quantitativ ely conﬁrms this geometric intuition relating ν ( Γ ) to the magnitude of d smallest eigenv alues of D − 1 1 L 1 (the random walk v ersion of the graph connection Laplacian): 1 d d X k =1 λ k  D − 1 1 L 1  ≤ ν ( Γ ) ≤ C d 3 λ 2 ( L 0 ) d X k =1 λ k  D − 1 1 L 1  , (50) where C > 0 is a constan t, λ 2 ( L 0 ) is the sp ectral gap of Γ asso ciated with the graph Laplacian L 0 , and λ k  D − 1 1 L 1  is the k th smallest eigen v alue of D − 1 1 L 1 . (The actual version stated in [13] is for the smallest d eigen v alues of the normalized graph connection Laplacian D − 1 / 2 1 L 1 D − 1 / 2 1 , but note that D − 1 / 2 1 L 1 D − 1 / 2 1 has the same eigenv alues as D − 1 1 L 1 .) Classical Cheeger inequalities [35, 3, 39] relate isop erimetric constants or cuts on graphs and manifolds to the sp ectral gap of a graph Laplacian or Laplace-Beltrami op erator. There hav e b een Cheeger-type inequalities for simplicial complexes with the ob jective of understanding high-dimensional generalization of expander graphs [117, 121, 122, 140]. These results are all concerned with partitioning graphs, manifolds, or simplicial complexes. The Cheeger-t yp e inequalit y in equation (50) diﬀers from standard Cheeger inequalities in that the co c hains are group- or vector-v alued. In addition, the frustration ν ( Γ ) is not related with any optimalily for graph partitioning — in the sense of Prop osition 2.1, ν ( Γ ) measures the triviality of a ﬁbre bundle as a whole. The algorithm we will prop ose in Section 3 is an attempt to address the graph cut problem based on the sync hronizabilit y of the partitions resulted. The Geometry of Synchronization Problems and Learning Group Actions 25 3 Learning Group Actions by Synchronization In this section w e specify an algorithm for learning group actions from observ ations based on sync hronization. W e also use sim ulations to provide some insigh t tow ards the p erformance of the algorithm. 3.1 Motiv ation and General F orm ulation W e ﬁrst state some basic terminology from the general theory of group actions that will b e used extensiv ely . If G is a group and X is a set, a left gr oup action of G on X is a map φ : G × X → X : ( g , x ) 7→ φ ( g , x ) suc h that φ ( e, x ) = x, ∀ x ∈ X if e is the identit y element of G and φ ( g , φ ( h, x )) = φ ( g h, x ) , ∀ x ∈ X, ∀ g , h ∈ G. T o simplify notation, we will abbreviate φ ( g , x ) as g .x . The orbit of any element x ∈ X under the action of G is deﬁned as the set G.x := { g .x | g ∈ G } . If w e introduce an equiv alence relation on X by setting x ∼ y ⇔ x = g .y for some g ∈ G, then clearly x ∼ y if and only if G.x = G.y . The set X is naturally partitioned into the disjoin t unions of orbits, and each orbit Y is an invariant subset of X under the action of G in the sense that G.Y ⊂ Y . If for an y pair of distinct elemen ts x, y of X there exists g ∈ G such that g .x = y , w e sa y that the action of G on X is tr ansitive . Note that the total space X is an inv ariant subset in its o wn right, and the action of G on eac h orbit is ob viously transitive. I f the set X is ﬁnite and there exists a constant time procedure to verify whether any tw o elements are equiv alen t under transformations, the problem of partitioning X into disjoint subsets of orbits can b e solv ed in p olynomial time complexity with respect to the size of X . In practice we are often in terested in classiﬁcation or clustering tasks which can be framed as follows: giv en a dataset X = { x 1 , · · · , x n } of n ob jects, ﬁnd a corresp ondence or transformation b et ween eac h pair of distinct ob jects. W e will see these pairwise correspondences often pla y the role of nuisance v ariables and one needs to “quotien t out” the inﬂuence of these v ariables in do wnstream analysis (e.g. for most practical applications of synchronization problems [138, 147, 10] and alignment problems in statistical shap e analysis [24]). The in tuition as to why some of these pairwise correspondences are nuisance v ariables one can often with greater ﬁdelity transform one ob ject into another via intermediary transformations to other ob jects rather than a direct transformation betw een ob jects. Sometimes, for instance in the analysis of a collection of shap es in computer graphics [84, 119, 82, 36, 91, 109] and group-wise registration in automated geometric morphometrics [23, 2, 102, 103, 94], the pairwise transformations con tains crucial information and are imp or- tan t on their own righ t. A common challenge in b oth of the ab ov e problems is that the ﬁdelity of pairwise comparisons can b e extremely v ariable ov er the data. W e illustrate this challenge using the example of com- puting con tin uous Pro crustes distances b et ween disk-type shapes [2] in automated geometric morphometrics. The core of the algorithm is an eﬃcien t strategy for searc hing the M¨ obius transformation group of the unit disk to obtain a diﬀeomorphism betw een the shap es that minimizes an energy functional. It has b een ob- serv ed that for similar shap es (in the sense of ha ving a small pairwise distance), the resulting diﬀeomorphism is often of high quality and can reﬂect the correspondence of biological traits. If the shape pair is highly dissimilar, the diﬀeomorphism tends to suﬀer from v arious structural errors (see e.g. [65] and [66, Chapter 5]). Similar issues hav e also b een observ ed in the ﬁeld of non-rigid shap e registration in geometric pro cessing — successful feature extraction and matc hing tec hniques for near-isometric shap es ab ound [143, 8, 27, 127, 95, 99], whereas registering shap e pairs with large deformation is still considered a diﬃcult op en problem [26, 96, 1]. Recently , a series of works [119, 84, 82, 36, 91, 109] prop osed to jointly compute all pairwise corre- sp ondences within a collection sub ject to “consistency constraints” that require the composition of resulted maps along an y cycle within the collection be appro ximately the iden tity map. The idea in this approach is that pairwise correspondences b et ween dissimilar shap es are implicitly appro ximated b y concatenating man y corresp ondences b etw een similar shap es with the individual correp ondences hav e high ﬁdelit y , thus av oiding directly solving non-conv ex optimization problems with large num b ers of lo cal minimizers. Similar ideas can also be found in recent progress in automated geometric morphometrics where a Minimum Sp anning T r e e 26 Tingran Gao, Jacek Bro dzki, and Say an Mukherjee (MST) provides the concatenating of corresp ondences [24, 151, 65]. It has b een observ ed by morphologists that cycle-consistent constrain ts are more often satisﬁed for a collection of samples within a sp ecies versus samples across a v ariety of species, suggesting that inconsistency ma y b e used for sp ecies clustering. Motiv ated b y the ab ov e algorithms and approac hes, we propose to study the follo wing general problem of L e arning Gr oup A ctions (LGA): Problem 3.1 (Learning Group Actions) Given a group G acting on a set X , sim ultaneously learn a new action of G on X and a partition of X into disjoin t subsets X 1 , · · · , X K , such that the new action is as close as p ossible to the giv en action and cycle-consisten t on each X i (1 ≤ i ≤ K ). The LGA problem can also be understo o d as a v ariant of the classical clustering problem, in which the coarse-graining is based on the cycle-consistency of group actions rather than pairwise similarit y or spatial conﬁguration of elements in the dataset. A solution of the LGA problem provides not only a partition of the input dataset but also cycle-consisten t group actions within each cluster. It is useful to notice that all group elemen ts implemented as pairwise actions within the same partition X i form a subgroup of G ; the LGA problem can thus also be considered as “learning” subgroups of a prescrib ed “ambien t group” that optimally ﬁt a given dataset X . In other w ords, b y solving an LGA problem w e iden tify the “correct” transformation group for a dataset, which in most practical situations are m uch more tigh tly adapted to the given data than the potentially massive group of all possible transformations G . Example 3.1 If the set X is a ve ctor sp ac e and we se ek a dir e ct sum de c omp osition X = L K i =1 X i inste ad of a p artition X = S K i =1 X i , the LGA pr oblem r e duc es to the se ar ch for al l irr e ducible G -subr epr esentations of X . Example 3.2 Consider a p oint set X = { x 1 , · · · , x n } e quipp e d with a lab eling map S : X → {± 1 } that assigns to e ach x i either value +1 or − 1 . We say x i has p ositiv e spin if S ( x i ) = 1 and has negativ e spin if S ( x i ) = − 1 . L et G = {± 1 } act on X tr ansitively as ( g j i , x i ) 7→ x j , g j i = S ( x j ) S ( x i ) . Supp ose the spin of e ach p oint in X (i.e. the lab el map S ) is unknown, but we have ful l ac c ess to the gr oup actions { g ij } , we c an r e c onstruct S — up to ﬂipping lab els ± 1 — by sp e ctr al clustering the dataset X , viewe d as vertic es of a c omplete gr aph Γ with weight w ij = g ij on the e dge c onne cting x i and x j . Under cir cumstanc es wher e some gr oup actions g j i ar e subje ct to a sign-ﬂip err or (noisy me asur ements), or/and the gr aph Γ is not c omplete (inc omplete me asur ements), sp e ctr al or semi-deﬁnite pr o gr amming r elaxation te chniques c an stil l b e use d to r e c over S up to p ermuting lab els ± 1 (se e e.g. [34]). With X and (p otential ly noisy and inc omplete) { g j i } as input, this sp e ctr al clustering example c an b e c onsider e d as an instanc e of LGA: the output c onsists of a p artition of X into p ositive/ne gative spin subsets, as wel l as the trivial sub gr oup { +1 } of G = {± 1 } acting in a cycle-c onsistent manner on b oth p artitions. Example 3.2 pro vides further motiv ation to consider a v ersion of LGA in the con text of sync hronization problems. W e are giv en a graph Γ = ( V , E ) and the data X , where the vertex set V is identiﬁed with observ ations in X and the edges in E represen ting pairwise relations b etw een elements of X . It is natural to consider a p artition of the gr aph Γ in this setup decomp osition of Γ in to connected subgraphs such that the v ertices of the subgraphs form a partition of the set of vertices of Γ . Problem 3.2 (Learning Group Actions b y Synchronization) Let Γ = ( V , E ) b e an undirected w eighted graph, G a top ological group, and ρ ∈ C 1 ( Γ ; G ) a giv en edge potential on Γ . F urthermore, assume the vertex set V is equipped with a cost function Cost G : G × G → [0 , ∞ ). Denote X K for all partitions of Γ into K nonempt y connected subgroups ( K ≤ n ) and ν ( S i ) = inf f ∈ C 0 ( Γ ; G ) X j,k ∈ S i w j k Cost G ( f j , ρ j k f k ) , vol ( S i ) = X j ∈ S i d j , 1 ≤ i ≤ K. The Geometry of Synchronization Problems and Learning Group Actions 27 Solv e the optimization problem min { S 1 , ··· ,S K }∈ X K max 1 ≤ i ≤ K ν ( S i ) min 1 ≤ i ≤ K v ol ( S i ) (51) and output an optimal partition { S 1 , · · · , S K } together with the minimizing f ∈ C 0 ( Γ ; G ). In the follo wing, we shall refer to Problem 3.2 as L e arning Gr oup A ctions by Synchr onization (LGAS). When G = O ( d ) or U ( d ) and Cost G is the squared F rob enius norm on d × d matrices, ν ( S i ) is clearly the frustration (49) of the subgraph of Γ with v ertices in S i , up to a m ultiplicativ e constan t dep ending only on Γ and dimension d . The minimizing v ertex potential deﬁnes a sync hronizable edge p oten tial on the en tire graph Γ , thus also gives rise to a cycle-consisten t action on each partition. Note that the ob jectiv e function (51) do es not accoun t for the discrepancy b et ween the realized sync hronizable edge potential and the original ρ on edges across partitions — intuitiv ely , solving Problem 3.2 amounts to forming partitions by economically “dropping out” appropriate edges in Γ to minimize the total frustration. 3.2 SynCut: A Heuristic Algorithm for Learning Group Actions by Sync hronization In this subsection, we will in vestigate Problem 3.2 (LGAS) in the context of O ( d )-synchronization problems, fo cusing on the simpler setting where K = 2. In this case, (51) simpliﬁes to min S ⊂ V max { ν ( S ) , ν ( S c ) } min { vol ( S ) , v ol ( S c ) } . Note that max { ν ( S ) , ν ( S c ) } ≤ ν ( S ) + ν ( S c ) ≤ 2 max { ν ( S ) , ν ( S c ) } , w e can thus consider — drawing an analogy with the standard approach of studying Cheeger num b ers through normalized cuts — the follo wing optimization problem closely related with (51): ξ Γ := min S ⊂ V ξ ( S ) := min S ⊂ V [ ν ( S ) + ν ( S c )]  1 v ol ( S ) + 1 v ol ( S c )  . (52) Recall from (49) that ξ Γ further simpliﬁes into ξ Γ = min S ⊂ V inf g ∈ C 0 ( Γ ; O ( d )) 1 2 d 1 v ol ( Γ ) X i,j ∈ V ( i,j ) / ∈ ∂ S w ij k g i − ρ ij g j k 2 F · v ol ( Γ ) v ol ( S ) v ol ( S c ) = min S ⊂ V inf g ∈ C 0 ( Γ ; O ( d )) 1 2 d 1 v ol ( S ) v ol ( S c ) X i,j ∈ V ( i,j ) / ∈ ∂ S w ij k g i − ρ ij g j k 2 F , (53) where ∂ S := { ( u, v ) ∈ E | u ∈ S, v ∈ S c or u ∈ S c , v ∈ S } . In other words, the goal of solving the optimization problem (53) is to low er the total frustration of the graph Γ by dropping out a minimum set of edges under the constraint that the residual graph consists of t wo connected components; this is equiv alent to say that w e seek a most economic graph cut in terms of reducing total frustration. T o simplify statemen ts, w e shall refer to k g i − ρ ij g j k 2 F as the frustr ation on e dge ( i, j ) ∈ E of vertex p oten tial g with resp ect to the edge p otential ρ , and call the collection of frustrations on all edges the e dge-wise frustr ations . The sum of all edge-wise frustrations will b e referred to as the total frustr ation . F orm ulation (53) motiv ates a greedy algorithm that alternates betw een minimizing graph cuts and v ertex p oten tials. W e shall refer to this algorithm as Synchr onization Cut , or SynCut for short; see Algorithm 1. W e describe the main steps in SynCut below: 28 Tingran Gao, Jacek Bro dzki, and Say an Mukherjee Step 1. Initialization: Input data include the weigh ted graph Γ = ( V , E , w ), edge potential ρ ∈ C 1 ( Γ ; G ), and parameters required for the sp ectral clustering subroutine plus termination conditions for the main lo op. Initialize iteration coun ter t = 0, and dynamic graph w eigh ts  to be the input graph w eigh ts w ; Step 2. Glob al Synchr onization: Synchronize the edge p otential ρ on the entire graph Γ with resp ect to edge w eigh ts. An y sync hronization algorithm can be used in this step, e.g. sp ectral relaxation [13, 34] or SDP relaxation [10, 134, 34, 12, 118]. Note that in this step the sync hronization is performed on Γ with dynamic w eigh ts  instead of the original w eigh ts w . Denote f ( t ) as the edge p otential on Γ at the edge p oten tial at the t -th iteration; Step 3. Sp e ctr al Clustering (First Pass): Up date dynamic weigh ts  based on the frustration of f ( t ) on each edge b y  ij = w ij exp  − 1 σ    f ( t ) i − ρ ij f ( t ) j    2 F  , where σ > 0 is the a v erage of all non-zero edge-wise frustrations then partition the vertex set V of graph Γ in to K clusters S 1 , · · · , S K using sp ectral clustering based on the up dated dynamic w eights  . The goal is to cut the graph Γ into more sync hronizable clusters; edges causing large frustration are assigned relatively smaller w eigh ts  ij to increase the chance of b eing cut. T o simplify notation, w e will also use S ` (1 ≤  ≤ K ) to denote the subgraph of Γ spanned b y the v ertices in S ` ; Step 4. L o c al Synchr onization: Sync hronize the edge potential ρ within each partition S ` , 1 ≤  ≤ K . If w e denote ρ | S ` ,  | S ` for the restrictions of ρ ,  to S ` , resp ectiv ely , then this step solves the sync hronization problem on w eighted graph ( S ` ,  | S ` ) for prescrib ed edge p otential ρ | S ` . Again, an y synchronization algorithm can b e used in this step. Denote g ( ` ) for the resulting v ertex p otential on S ` ; Step 5. Col lage: After obtaining g ( ` ) for eac h lo cal synchronization on S ` , w e make a “collage” from these lo cal solutions to form a global vertex p otential deﬁned on the entire graph Γ . Since each g ( ` ) is obtained from synchronizing within S ` , the collected lo cal solutions  g ( ` )  K ` =1 generally incur large incompatibilit y (frustration) on edges across partitions. Our strategy is to ﬁnd K elements h 1 , · · · , h K ∈ G , where eac h h ` acts on g ( ` ) b y g ( ` ) u 7→ g ( ` ) u h ` , ∀ u ∈ S ` , suc h that the total cr oss-p artition frustr ation C  h 1 , · · · , h K   { S ` } 1 ≤ ` ≤ K , n g ( ` ) o 1 ≤ ` ≤ K  := X 1 ≤ p 6 = q ≤ K X ( u,v ) ∈ E u ∈ S p ,v ∈ S q w uv    g ( p ) u h p − ρ uv g ( q ) v h q    2 F (54) is minimized. Note that this is essentially synchronizing an edge p otential e ρ pq =        X ( u,v ) ∈ E u ∈ S p ,v ∈ S q w uv  g ( p ) u  − 1 ρ uv g ( q ) v if partitions S p , S q are connected 0 otherwise (55) on a reduced complete graph e Γ K consisting of K v ertices where each v ertex represent one of the K parti- tions S 1 , · · · , S K . It thus simply requires calling the synchronization routine again to obtain h 1 , · · · , h K , but this time the scale of the synchronization problem is often m uch smaller than the previous global and lo cal synchronization steps. Also note that for the binary cut case K = 2 and G = O ( d ) this collage step is even simpler: it suﬃces to p erform an single SVD on the d × d matrix X ( u,v ) ∈ E u ∈ S 1 ,v ∈ S 2 w uv  g ( p ) u  − 1 ρ uv g ( q ) v = U Σ V > and set h 1 = U V > , h 2 = I d × d . Step 6. Sp e ctr al Clustering (Se c ond Pass): Up date dynamic weigh ts  based on the frustration of f ∗ on eac h edge by  ij = w ij exp  − 1 σ   f ∗ i − ρ ij f ∗ j   2 F  , where σ > 0 is the a v erage of all non-zero edge-wise frustrations then partition Γ in to K clusters S 1 , · · · , S K using spectral clustering for a second time, based on the up dated dynamic w eights  . The Geometry of Synchronization Problems and Learning Group Actions 29 Step 7. R ep e at Step 2 to Step 6 Until Conver genc e. The termination condition can be speciﬁed either by a maxim um n umber of iterations or monitoring the change of the quan tit y ξ ( { S 1 , · · · , S K } ) := K X ` =1 ν ( S ` ) ! K X k =1 1 v ol ( S k ) ! . (56) A t the end of the pro cedure, return the partitions { S 1 , · · · , S K } and the ﬁnal edge p otential f ∗ from the most recen t updates. The cycle-consisten t edge p otential on partition S ` is encoded in the restriction of f ∗ to S ` . Algorithm 1 Synchroniza tion Cut : Learning Group Actions by Synchronization 1: pro cedure SynCut ( Γ , ρ , K ) . weigh ted graph Γ = ( V , E , w ), ρ ∈ C 1 ( Γ ; G ), num b er of partitions K 2: t = 0 3:  = w 4: while not conv erge do 5: f ( t ) ∈ C 0 ( Γ ; G ) ← Synchronize ( Γ, ρ,  ) 6: σ ← av erage non-zero edge-wise frustrations of f ( t ) 7: for ( i, j ) ∈ E do . calculate weigh ts  on graph Γ for sp ectral clustering 8:  ij ← w ij exp  − 1 σ    f ( t ) i − ρ ij f ( t ) j    2 F  9: end for 10: { S 1 , · · · , S K } ← SpectralClustering ( Γ,  ) 11: for ` = 1 , · · · , K do 12: g ( ` ) ∈ Ω 0 ( S ` ; G ) ← Synchronize  S ` , ρ | S ` ,  | S `  13: end for 14: f ∗ ∈ Ω 0 ( Γ ; G ) ← Collage  { S ` } K ` =1 ,  g ( ` )  K ` =1  15: σ ← av erage non-zero edge-wise frustrations of f ∗ 16: for ( i, j ) ∈ E do . up date w eights  on graph Γ for next iteration 17:  ij ← w ij exp  − 1 σ   f ∗ i − ρ ij f ∗ j   2 F  18: end for 19: { S 1 , · · · , S K } ← SpectralClustering ( Γ,  ) 20: t ← t + 1 21: end while 22: return { S 1 , · · · , S K } , f ∗ . f ∗ deﬁnes a cycle-consistent edge p otential on eac h partition 23: end pro cedure 3.3 Results on Sim ulated Random Synchronization Net works In this subsection, we use simulations to provide some in tuition for the b eha vior of SynCut under the setting K = 2 (tw o partitions). W e ﬁrst specify a random pro cedure to sim ulate input data — a connected random graph with a prescrib ed edge p otential — for sync hronization problems. In addition, the random graph generation pro cedure will b e controlled by a parameter that allows us to adjust the lev el of obstruction to the synchronizabilit y of the prescrib ed edge p otential ov er the generated graph. W e then sp ecify the metrics used for p erformance measure. W e conclude by demonstrating that the partition generated from SynCut recov ers the t wo sync hronizable connected comp onents with high accuracy and within relatively few n umbers of iterations. F or the simplicit y of statements, we refer to eac h pair of generated graph and edge p oten tial an instance of a r andom synchr onization network . 3.3.1 R andom Synchr onization Network Simulation W e ﬁrst specify the pro cedure to generate the random graphs. Our in ten tion is to sample random graphs with suﬃcien tly v ariable sp ectral gaps, based on the intuition that a large sp ectral gap of the underlying graph results in greater obstruction to the sync hronizability of the edge p otential constructed by the pro cedures that will soon be described in this subsection. W e ﬁrst generate tw o partitions S 1 , S 2 with an equal n umber 30 Tingran Gao, Jacek Bro dzki, and Say an Mukherjee of v ertices. Each partition is a connected comp onent built from a vertex degree sequence of random integers uniformly distributed in an interv al (sa y 5 to 8), adapting an algorithm ﬁrst prop osed in [17]; when the in terv al is a single integer, the connected comp onen t is a regular graph. Random edges are than added to link the tw o partitions S 1 , S 2 . The num ber of inter-component random edges p ositiv ely correlates with the sp ectral gap, as sho wn in Figure 1, suggesting that this num b er can b e used as a parameter to adjust the lev el of obstruction to cutting the graph into tw o connected comp onen ts S 1 and S 2 . A subtlety in this random netw ork generation pro cedure is that a uniform distribution on the num b er of in ter-comp onent links does not induce a uniform distribution on the spectral gaps of the generated random graphs, due to concentration eﬀects. A precise c haracterization of the distribution of spectral gaps in our random graph mo del is in teresting on its o wn righ t but b eyond the scop e of this pap er. W e refer in terested readers to the existing literature on the sp ectral gaps of random graphs such as [40, 44, 81]. In practice, we simply use a large n um b er of random trials to generate suﬃciently man y sample graphs with spectral gaps within desirable ranges; see Figure 2a. After dra wing an instance of the random graph, w e randomly construct an edge p oten tial that is sync hro- nizable within S 1 and S 2 , but not necessarily synchronizable on the inter-component links. The pro cedure to generate the random edge potentials pro ceeds as follows: (1) Randomly generate a v ertex potential g ∈ C 0 ( Γ ; G ) for the en tire graph Γ ; (2) Set the v alue of ρ on edge ( i, j ) according to ρ ij = ( g i g − 1 j if both i, j ∈ S 1 or i, j ∈ S 2 , a random matrix in O ( d ) otherwise. The v ertex potential g ∈ C 0 ( Γ ; G ) will no longer attain the minim um frustration for the entire graph with resp ect to the prescrib ed edge p otential ρ , due to the edges added b etw een the partitions that are muc h less lik ely sync hronizable. Fig. 1 A scatter plot displaying the correlation betw een the num b er of inter-component links and the sp ectral gap in our random graph mo del, with N = 100 vertices and the (integer) num b er of inter-component links uniformly distributed b etw een 100 and 250. W e consider each run of SynCut as successful if b oth output partitions are synchronizable connected comp onen ts, i.e. if SynCut reco vers the original partitions S 1 , S 2 . The p erformance of SynCut is measured using the err or r atio computed by dividing the num ber of erroneously clustered vertices b y the total num b er of vertices. If SynCut successfully recov ers S 1 , S 2 , this error ratio is 0; if the output partition is close to a random guess, or if the algorithm fails to separate the v ertices in to distinct clusters, the error ratio is 0 . 5. The error ratios of the partitions output from SynCut are then compared with a baseline graph cutting algorithm using normalized graph cut (NCut) that do es not utilize an y information in the prescrib ed edge potential; see e.g. [129, 152]. W e refer interested readers to [9] for an initial attempt at analyzing this phenomenon for The Geometry of Synchronization Problems and Learning Group Actions 31 the scenario where the group G is a symmetric group (group of p erm utations) and the underlying graph is generated from a stochastic blo ck mo del. 3.3.2 Simulation R esults Fig. 2 (a) Histogram of the sp ectral gap of the 10 , 000 random graphs drawn from our mo del. (b) Histogram of the error ratios of the SynCut clustering results. (c) Histogram of the error ratios of the baseline NCut clustering results. (d) Histogram of the num ber of iterations for SynCut. (e) Scatter plots of the error ratios of SynCut and NCut versus spectral gap. In this simulation study , w e set the n um b er of vertices in eac h of the t wo synchronizable comp onents to N = 100, and the en tries of the v ertex degree sequence of random in tegers are independently uniformly dis- tributed b et ween 4 and 8. The num ber of inter-component links b et ween the t wo sync hronizable comp onen ts is drawn uniformly b etw een 100 and 250. The edge potentials are v alued in the orthogonal group O ( d ) with d = 5. W e terminate SynCut either after 10 iterations or if the c hange in the v alue of the ob jectiv e function ξ [see (56)] b etw een consecutive iterations falls b elow a preset tolerance of 10 − 8 . W e plot in Figure 2(a) the sp ectral gaps of 10 , 000 realizations of our random netw ork mo del. In Figure 2(b) and Figure 2(c) w e observ ed that the error ratios in these 10 , 000 runs of SynCut tend to b e m uch smaller than NCut, suggesting that SynCut outputs more accurate partitions with resp ect to synchronizabilit y . In Figure 2(e), w e again see that SynCut outp erforms NCut and the amount of improv emen t increases with the magnitude of the sp ectral gap. Figure 2(d) shows that SynCut conv erges quickly . W e no w focus on a particular instance of a random synchronization net work to better understand SynCut in comparison with the sp ectral relaxation algorithm prop osed in [13]. Each synchronizable comp onen t in the random sync hronization netw ork shown in Figure 3(a) is a regular graph containing N = 100 vertices 32 Tingran Gao, Jacek Bro dzki, and Say an Mukherjee Fig. 3 (a) A random graph consisting of tw o synchronizable connected comp onents, eac h with 100 vertices and 250 edges (in blue), and 100 non-sync hronizable in ter-comp onent edges (in red). The edge potential tak es v alue in the orthogonal group O (5). All vertex degrees in each synchronizable comp onen t are set to 5. (b) Edge-wise frustration for the vertex p otential g used to generate the prescrib ed edge p otential ρ . As exp ected, frustration is small within each connected component but large b etw een components. (c) Edge-wise frustration for the vertex p otential obtained from sp ectral relaxation [13]. The total frustration is low er than that in the top right ﬁgure, but the inter-component edges carries relatively low er frustration since the relaxation procedure tends to “spread” the non-synchronizabilit y across the en tire graph. (d) Edge-wise frustration for the v ertex potential obtained from SyncCut. The total frustration is higher than that for the sp ectral relaxation solution, but the distribution of frustrations on the edges is closer to that of vertex g and can thus b e used to recov er the synchronizable connected comp onents. and 250 edges, generated with a constan t v ertex degree sequence of 5. W e color the edges within and betw een sync hronizable components in blue and red, respectively . In Figure 3(b) we plot the edge-wise frustration for the vertex p otential g used to generate the edge p otential ρ prescrib ed to the netw ork. As exp ected, the frustration is zero within each sync hronizable comp onent but large on the edges across comp onen ts. Figure 3(c) and Figure 3(d) show the edge-wise frustrations for tw o vertex p oten tials obtained from the sp ectral relaxation algorithm [13] and SynCut, respectively . Though the total frustration is larger for SynCut than sp ectral relaxation, the SynCut solution concen trates most of the frustration on the non-sync hronizable in ter-comp onent edges, with a distribution of edge-wise frustrations closer to the distribution for the initial v ertex p oten tial g . This suggests that applying a sp ectral graph cut algorithm using the edge-wise frustration of the SynCut solution as a dissimilarit y measure is adv antageous, as the distribution in Figure 3(d) iden tiﬁes the obstructions to sync hronizabilit y in the sync hronization net w ork more accurately . 4 Application to Automated Geometric Morphometrics In this section we formulate a problem in automated geometric morphometrics in terms of LGAS, then apply the SynCut algorithm to provide a solution. In Section 4.1 w e pro vide some bac kground in geometric morphometrics and its relation to synchronization problems. In Section 4.2 w e apply SynCut to a collection of second mandibular molars of prosimian primates and non-primate close relativ es. The morphological h yp othesis is that the geometric traits of second mandibular molars cluster in to 3 dietary regimens: folivor ous (herbiv ores that eat leav es), frugivor ous (herbivores or omnivores that prefer fruit), and inse ctivor ous (a carniv ore that eats insects). W e will show the SynCut result, which is based on the synchronizabilit y of pairwise correspondences, and compare it with a distance-based clustering result using diﬀusion maps [41]. The Geometry of Synchronization Problems and Learning Group Actions 33 4.1 Geometric Morphometrics and Synchronization The classic to ol in geometric morphometrics is Pr o crustes analysis . The basic assumption underlying this analysis framework is that most of the geometric information on each shap e can b e eﬃcien tly enco ded in a set of landmark p oints carefully pic ked to highligh t the morphometrical phenotypes (v ariation in the geometric shap e of an organism). The Pr o crustes distanc e b etw een tw o shap es is the av erage Euclidean distance betw een corresp onding landmarks, after applying a rigid motion (rotations, reﬂections, translations, and their comp ositions) to optimally align the tw o sets of landmarks. If all the shap es are marked with an equal num b er of landmarks but the landmark correspondence is not kno wn a priori , a combinatorial search can b e p erformed ov er all permutations of one-to-one landmark corresp ondences, and the minimum av erage Euclidean distance b etw een corresp onding landmarks can b e taken as a dissimilar measure betw een the tw o shap es. Comparing a pair of shap es in this framew ork th us yields abundant pairwise information, including a scalar dissimilarity score, a rigid motion, and a p erm utation matrix enco ding the one-to-one landmark corresp ondence. In automated geometric m orphometrics, landmark p oin ts are not used to represent the shap es, and al- gorithms search for an “optimal” transformation b etw een a pair of whole shap es directly by minimizing energy functionals o ver a set of admissible transformations. Dep ending on the sp eciﬁc class of transforma- tions and energy functional, the pairwise comparisons pro duce diﬀerent types of correspondences betw een surfaces, such as conformal/quasiconformal transformations, isometries, area-preserving diﬀeomorphisms, or ev en transp ort-plans b et ween surface area measures in a W asserstein framework. Regardless of the type of admissible transformations, the algorithm can output a rigid motion for the optimal alignment b et ween tw o shap es, as well as a dissimilarity or similarity score for such an alignment. See Figure 4 for an example of represen ting a collection of shapes using landmarks v ersus triangular meshes. Fig. 4 The meshes of four lemur molars from an an anatomical surface dataset ﬁrst published in [ 23]. The colored dots on the molars are landmark p oints where identical colors indicate corresp onding landmarks. When the analysis is extended from comparing a single pair to a large collection of shap es, a crucial premise for downstream statistical analysis (e.g. Gener al Pr o crustes Analysis (GP A) [74, 55, 75]) is that the pairwise corresp ondences b e cycle-c onsistent , meaning that propagating any landmark on any shap e by consecutiv e corresp ondences along a close cycle of shap es should land exactly at the original landmark. T ra- ditional landmark-based Pro crustes analysis b egins with consistently picking an equal num b er of landmarks on each shap e, resulting in a large amoun t of pairwise corresp ondence relations that are cycle-consistent by construction. This is, ho wev er, not the situation with automated geometric morphometrics, where the cor- resp ondence transformations pro duced b y automated algorithms are rarely cycle-consisten t, ev en when one lo calizes the transformations within relatively “stable” regions where landmarks are aﬃxed with the knowl- edge of an experienced geometric morphometrician. The necessity of cycle-consisten t correspondences links automated geometric morphometrics to synchronization problems. An automated geometric morphometric algorithm will output for eac h pair of shap es a triplet consisting of a dissimilarit y score, a rigid motion, and a pairwise transformation. W e can use the dissimilarit y scores to deﬁne a w eigh ted graph Γ that captures the similarities within the collection, b oth qualitativ ely and quantitativ ely , by adjusting the n umber of nearest neigh b ors of eac h v ertex and the w eights on eac h edge. The rigid motions and pairwise transformations deﬁne 34 Tingran Gao, Jacek Bro dzki, and Say an Mukherjee t wo edge p otentials on Γ , taking v alues in diﬀeren t groups. W e list below some in teresting sync hronization problems arising from this formulation: Three-Dimensional Euclidean Group. The rigid motions R ij b et ween shapes S i and S j that share an edge in Γ deﬁne an edge p otential R ∈ C 1 ( Γ ; E (3)), where E (3) is the three-dimensional Euclidean group. Solving an E (3)-sync hronization problem ov er Γ with respect to R results in a globally consistent alignmen t for a collection of shap es, which is often crucial for initializing geometric morphometrical analysis algorithms such as Dirichlet Normal Ener gy [31], Orientation Patch Analysis [60], and R elief Index [22]. Algorithms that automatically align a collection of anatomical shap es in a globally consistent manner can also be view ed as primitive approaches for solving E (3)-sync hronization problems; see e.g. [124, 24, 120, 73]. Orthogonal Group and Orien tability Detection. If the shap es are prepro cessed to sup erimp ose the cen ters of mass at the same point, the translation comp onen t of eac h R ij output from a pairwise landmark- based Pro crustes analysis v anishes 3 . This reduces the global alignmen t problem to standard synchronization problems ov er the compact Lie group O (3). Sp ectral and semideﬁnite programming (SDP) relaxation meth- o ds can then b e applied directly to solve the global alignment problem. If w e consider the edge p otential ρ ∈ C 1 ( Γ ; Z 2 ) deﬁned by ρ ij = det R ij , a Z 2 -sync hronization solution can b e used to either partition the dataset in to “left-handed” and “righ t-handed” subsets or conclude that suc h an orientation-based partition do es not exist. W e stated a similar situations in Example 3.2; other examples in this setting can b e found in applications of Orientable Diﬀusion Maps [135]. Automorphism Groups. Certain classes of transformations C ij b et ween eac h pair of shap es S i , S j giv e rise to an edge p otential on the graph Γ v alued in an automorphism group of a canonical domain. F or instance, algorithms such as M¨ obius V oting and the Continuous Pr o crustes Distanc e [2] betw een disk-type surfaces rely on the computation of conformal maps b etw een tw o shap es, based on uniformization parametrization tec hniques [123, 4] that map eac h surface conformally to a canonical unit disk on the plane. By in tert wining C ij with the parametrizations of the source and the target shap e, the corresp ondence b et ween S i and S j can b e equiv alen tly considered as an element of the conformal automorphism group Aut ( D ) of the planar unit disk D . The group Aut ( D ) is isomorphic to the pr oje ctive sp e cial line ar gr oup PSL (2 , R ), a non-compact simple real Lie group that is equiv alent to the quotient of the sp e cial line ar gr oup SL (2 , R ) b y {± I 2 } , where I 2 denotes the 2 × 2 iden tit y matrix. Sync hronization problems o v er PSL (2 , R ) or SL (2 , R ) require non-trivial extensions of the non-unique games (NUG) framework [11] o ver compact Lie groups. Group oids. Other types of transformations C ij b et ween each pair of shapes S i , S j require further gen- eralizations of the sync hronization framework to edge p otentials taking v alues in a gr oup oid rather than in a group. As an example, consider surface registration tec hniques based on area-preserving maps [2, 160, 159, 142]. These techniques use conformal or area-preserving parametrizations to push forward surface area mea- sures on S i , S j to measures µ i , µ j on the planar unit disk D , resp ectiv ely , then solv e for a tr ansp ort map on D that pulls bac k µ j to µ i (or equiv alently µ i to µ j ). T o form ulate suc h “transport-map-v alued” edge potentials in a synchronization framework, an edge p otential should be allow ed to take v alues in diﬀeren t classes of maps on diﬀeren t edges, with the only constrain t that maps on consecutiv e edges can b e comp osed; these ingredien ts hav e m uch in common in spirit with fundamental gr oup oids [29, 28] and Haeﬂiger’s c omplexes of gr oups [76, 77]. Suc h a generalized framework for synchronization problems can also b e used to analyze corresp ondences { C ij } that are soft maps [125, 139] or tr ansp ort plans [102, 103, 99], where one replaces the set of transport maps betw een µ i and µ j with (probabilistic) couplings Π ( µ i , µ j ) as in Kantoro vich’s relax- ation to the Monge optimal transp ort problem [149, 150]. The Horizontal Diﬀusion Maps (HDM) framew ork [67] and the application in automated geometric morphometrics [66] are among the initial attempts in this direction. 4.2 Clustering Lemurs by Dietary Regimens using Sync hronizabilit y of Molar Surfaces W e fo cus on a real anatomical surface mesh dataset of second mandibular molars from 5 genera of prosimian primates and nonprimate close relatives. There are a total of 50 molars with 10 sp ecimens from each genus; see Figure 5. The ﬁve genera divide in to three dietary regimens: the Alouatta and Br achyteles are folivo- rous, the Ateles and Cal lic ebus are frugivorous, and the Saimiri are insectiv orous. In Figure 4 we displa y 3 Note this is not the case for join tly analyzing a collection of shap es in a landmark-based Pro crustes analysis framew ork; see e.g. [34]. The Geometry of Synchronization Problems and Learning Group Actions 35 four lemur molars from an anatomical surface dataset ﬁrst published in [23], together with landmarks on eac h molar placed b y ev olutionary an thropologists. Similar datasets ha v e b een studied in a series of papers dev eloping algorithms for automatic geometric morphometrics [23, 2, 102, 103, 94, 69, 68]. The chewing surface of eac h molar is digitized as a tw o-dimensional triangular mesh in R 3 of disk-t yp e top ology (i.e. conformally equiv alen t with a planar disk). W e will apply SynCut to these 50 molars and examine if the clustering is consisten t with dietary regimens. Fig. 5 Consistent alignment of 50 lemur teeth based on applying SynCut to all pairwise alignments from the contin uous Pro- crustes analysis [2]. Each row corresp onds to teeth from a genus, from top to bottom: Alouatta , Ateles , Br achyteles , Cal lic ebus , Saimiri . Metho d W e ﬁrst pre-pro cess the dataset by translating and scaling each shap e so that all surface meshes cen ter at the origin and enclose unit surface area. W e then apply the contin uous Pro crustes distance algorithm for each pair of teeth, generating a a distance score d ij , a diﬀeomorphism C ij , and an orthogonal matrix R ij ∈ O (3) that optimally aligns S j to S i with resp ect to the diﬀeomorphism C ij . W e use the distance scores to construct a weigh ted K -nearest-neighbor graph Γ . The weigh ts are deﬁned as w ij = exp  − d 2 ij . σ 2  with the bandwidth parameter σ > 0 set to b e of the order of the a verage smallest non-zero distances. W e apply SynCut to the edge potential ρ ∈ C 1 ( Γ ; O (3)) deﬁned b y the alignmen ts R ij on the w eighted graph ( Γ , w ). Finally , we compare the clustering p erformance of SynCut with applying diﬀusion maps and sp ectral clustering directly to the weigh ted graph without the alignment information. R esults SynCut and diﬀusion maps b oth require the choice of a parameter K determining the num ber of nearest neighbors in the construction of the graph Γ . When 6 ≤ K ≤ 10 b oth procedures accurately cluster the 50 molars in the dataset into the three distinct dietary regimens, see Figure 6 for the t wo-dimensional em b edding plots for K = 7. SynCut pro duces sligh tly tighter and more distinguishable sp ecies clusters. Not surprisingly , for K > 10 — when the n um b er of nearest neigh bors exceeds the n umber of sp ecimens in each gen us — b oth algorithms are less accurate as K increases, with the accuracy of SynCut dropping faster than 36 Tingran Gao, Jacek Bro dzki, and Say an Mukherjee diﬀusion maps. This empirical observ ation is consistent with our in tuition that the performance of SynCut is more sensitive to increased sp ectral gaps than diﬀusion maps. Fig. 6 Embeddings of the 50 lemur teeth dataset into R 2 , obtained by applying diﬀusion maps (left) and SynCut (right) to the 7-nearest-neighbor graph. Both plots are p ost-processed using t-SNE [105]. (a) Diﬀusion maps applied to the weigh ted graph ( Γ, w ) successfully distinguishes three diet groups, but the genera are less distinguishable. (b) SynCut pro duces an edge-wise frustration matrix after the ﬁnal iteration that can be used by diﬀusion maps to generate a low-dimensional embedding, in which b oth dietary groups and genera are more distinguishable. 5 Conclusion and Discussion W e provided in this pap er a geometric framework for synchronization problems. W e ﬁrst related the syn- c hronizability of an edge potential on a connected graph to the trivialit y of a ﬂat principal bundle o v er the top ological space underlying the graph, then characterized sync hronizabilit y from tw o aspects: the holonomy of the principal bundle, and the t wisted cohomology of an asso ciated v ector bundle. On the holonom y side, w e established a corresp ondence b etw een t wo seemingly distan t ob jects on a connected graph Γ , namely , the orbit space of the action of G -v alued vertex p otentials on G -v alued edge p otentials, and the representation v ariet y of the fundamental group of Γ into G ; on the cohomology side, w e built a twisted de Rham co chain complex on an asso ciated vector bundle B ρ [ F ] of the synchronization principal bundle B ρ , of which the zero-th degree cohomology group characterizes the obstruction to the synchronizabilit y of the prescrib ed edge potential. With the presence of a metric on the asso ciated v ector bundle B ρ [ F ], we also developed a twisted Ho dge theory on graphs. Indep enden t of the con tribution to sync hronization problems, this theory is both a discrete v ersion of the Ho dge theory of elliptic complexes and a ﬁbre bundle analogue of the discrete Hodge theory on graphs. Speciﬁcally for synchronization problems, this twisted Ho dge theory realizes the graph connection Laplacian op erator as the zero-th degree Ho dge Laplacian in the twisted de Rham co chain complex. A Ho dge-t yp e decomp osition theorem is also pro v en, stating that the image of the twisted co diﬀeren tial is the orthogonal complement of the linear space of F -v alued synchronization solutions, with resp ect to the bundle metric. Motiv ated by the geometric in tuitions gained from these theoretical results, we coined the problem of learning group actions (LGA), and prop osed a heuristic algorithm, which w e referred to as SynCut, based on iteratively applying synchronization and sp ectral graph tec hniques. Numerical sim ulations on synthetic and real datasets indicated that SynCut has the potential to cluster a collection of ob jects according to the sync hronizability of a subset of partially observ ed pairwise transformations. The Geometry of Synchronization Problems and Learning Group Actions 37 W e conclude this pap er by listing several problems of interest for future exploration. These are only a subset of a v ast collection of p otential directions concerning the mathematical, statistical, and algorithmic asp ects of sync hronization problems: 1) The R epr esentation V ariety of Synchr onization Pr oblems. When a prescrib ed edge p otential ρ is not sync hronizable o ver graph Γ , the goal of the synchronization problem is to ﬁnd a sync hronizable edge p oten tial ˜ ρ that is as close as p ossible to ρ in a sense that has b een made clear in this pap er. The p oint of view adopted in Section 2.1 is that the sync hronization problem essentially concerns the orbits of ρ and ˜ ρ under the action of all vertex p oten tials. It is natural to conceive a synchronization algorithm based on the geometry of the orbit space C 1 ( Γ ; G ) /C 0 ( Γ ; G ) that enables eﬃciently “moving across” the orbits. Since the fundamen tal group of any connected graph is simply a free pro duct of copies of Z , we exp ect the represen tation v ariety Hom ( π 1 ( Γ ) , G ) /G to possess relatively simple structures that could be used for guiding the design of no vel sync hronization algorithms with pro v able guarantees. 2) Higher-or der Synchr onization Pr oblems. As a simplicial complex, the graph Γ only has 0- and 1-simplices, whic h results in only one cohomology group of interest in the de Rham co c hain complex (41). By extend- ing the t wisted de Rham and Ho dge theory developed in Section 2.2 to simplicial complexes of higher dimensions, w e exp ect higher-order synchronization problems can be form ulated and studied using to ols and insights from high-dimensional expanders and the Ho dge theory of elliptic complexes. Generalizing the current regime of synchronization problems, in which only pairwise transformations are considered, the higher-order analogies would enable the study of relations and in teractions among m ultiple vertices in the graph Γ , which p otentially op ens do ors tow ards higher-order graphical mo dels and related statistical inference questions as w ell. 3) Hier ar chic al Partial Synchr onization A lgorithms with Pr ovable Guar ante es. The SynCut algorithm we prop osed in this pap er can be understo o d as an iterative hierarc hical partial sync hronization algorithm, based on the assumption that edge-wise synchronization is an indicator of the sync hronizability of a prescrib ed edge p oten tial o ver a prop er subgraph. The numerical exp eriments on syn thetic and real datasets suggested the v alidity of this in tuition under our random graph mo del, but no prov able guaran tees exist either for the con vergence or the eﬀectiveness of algorithms similar or related to SynCut, to the b est of our knowledge. Building a Cheeger-type inequality as the p erformance guarantee for SynCut attracted our attention, but ev en the analogy of Cheeger num ber (or graph conductance) in the setting of SynCut or LGA is not clear — whereas the Cheeger num ber dep ends only on the graph weigh ts, whic h are ﬁxed num b ers on each edge indep endent of the graph cut, the notion of edge-wise frustration is highly non-lo cal as the frustration depends on the b ehavior of the sync hronization solution on the entire graph. W e conjecture that a Cheeger-type inequality for SynCut, if exists, will reﬂect the global geometry information encoded b y geometric quantities asso ciated with the ﬁbre bundle. 4) Statistic al F r amework for L e arning Gr oup A ctions. The LGA problem presented in this pap er is not form ulated with a natural generative mo del for the dataset of ob jects with pairwise transformations; nor is assumed any concrete noise mo dels. It w ould b e of interest to provide a systematic, statistical framew ork under whic h the problem of LGA and LGAS can b e quantitativ ely analyzed and understo o d; w e believe such a framework also has the p otential to bridge statistical inference with sync hronization problems. App endix A Pro ofs of Prop osition 1.1 and F orm ula (37) Pr o of (Pr o of of Pr op osition 1.1) The construction of U using the stars of the v ertices of Γ ensures that (1) U i ∩ U j 6 = ∅ if and only if ( i, j ) ∈ E ; (2) U i ∩ U j ∩ U k 6 = ∅ if and only if the 2-simplex ( i, j, k ) is in X . F or suc h pair ( i, j ), deﬁne constan t map g ij : U i ∩ U j → G as g ij ( x ) = ρ ij ∀ x ∈ U i ∩ U j . Set g ii = e for all 1 ≤ i ≤ | V | , and note that g ij ( x ) = g − 1 j i ( x ) for all x ∈ U i ∩ U j b y our assumption on ρ . If ρ is synchronizable ov er G , let f : V → G b e a vertex p otential satisfying ρ , then ρ ij = f i f − 1 j for all ( i, j ) ∈ E 38 Tingran Gao, Jacek Bro dzki, and Say an Mukherjee from (1). Thus ρ kj ρ j i = ρ ki for any triangle ( i, j, k ) in Γ , or equiv alen tly that g kj ( x ) g j i ( x ) = g ki ( x ) for all x ∈ U i ∩ U j ∩ U k . Therefore, { g ij | 1 ≤ i, j ≤ | V |} deﬁnes a system of c o or dinate tr ansformations [141, § 2] with v alues in G . These data determine a principal ﬁbre bundle P ρ with base space X and structure group G — by a standard construction in the theory of ﬁbre bundles (see e.g. [141, § 3.2]) — of whic h local trivializations are deﬁned on the op en sets in U with constan t transition functions g ij ; this principal bundle is thus ﬂat by deﬁnition. F urthermore, the vertex p oten tial f and the compatibilit y constrain ts (1) ensure that the following global section s : X → P ρ is w ell-deﬁned on this bundle: s ( x ) = φ i ( x, f i ) , x ∈ U i where φ i : U i × G → P ρ is the lo cal trivialization of P ρ o ver U i . The triviality of this principal bundle then follo ws from the existence of suc h a global section; see e.g. [141, § 8.3]. The other direction of the prop osition follo ws immediately from this trivialit y criterion for principal bundles. Pr o of (Pr o of of F ormula (37) ) h ω , η i = 1 2 X ( i,j ) ∈ E h w ij D p i  ω ( i ) ij  , p i  η ( i ) ij E F + w j i D p j  ω ( j ) j i  , p j  η ( j ) j i E F i = 1 2 X ( i,j ) ∈ E h w ij D p i  ω ( i ) ij  , p i  η ( i ) ij E F + w ij D ρ ij p j  ω ( j ) j i  , ρ ij p j  η ( j ) j i E F i = 1 2 X ( i,j ) ∈ E h w ij D p i  ω ( i ) ij  , p i  η ( i ) ij E F + w ij D p i  ω ( i ) j i  , ρ ij p i  η ( i ) j i E F i (see compatibilit y condition (31)) = 1 2 X ( i,j ) ∈ E h w ij D p i  ω ( i ) ij  , p i  η ( i ) ij E F + w ij D p i  ω ( i ) ij  , ρ ij p i  η ( i ) ij E F i (sk ew-symmetry) = X ( i,j ) ∈ E w ij D p i  ω ( i ) ij  , p i  η ( i ) ij E F . App endix B Graph Laplacian in Discrete Ho dge Theory Deﬁne K -v alued 0- and 1-forms on w eighted graph Γ = ( V , E , w ) as Ω 0 ( Γ ) := { f : V → K } , Ω 1 ( Γ ) := { ω : E → K | ω ij = − ω j i ∀ ( i, j ) ∈ E } , equipp ed with natural inner pro ducts h f , g i := X i d i h f i , g i i K , ∀ f , g ∈ Ω 0 ( Γ ) , h ω , η i := X ( i,j ) ∈ E w ij h ω ij , η ij i K , ∀ ω , η ∈ Ω 1 ( Γ ) , where h· , ·i K is an inner pro duct on K , and d i = P j :( i,j ) ∈ E w ij is the weigh ted degree at v ertex i ∈ V . Analogous to the study of diﬀerential forms on a smo oth manifold, one can deﬁne the diﬀer ential d : Ω 0 ( Γ ) → Ω 1 ( Γ ) and c o diﬀer ential δ : Ω 1 ( Γ ) → Ω 0 ( Γ ) operators that are formal adjoints of eac h other: ( d f ) ij = f i − f j , ∀ f ∈ Ω 0 ( Γ ) , ( δ ω ) i := 1 d i X j :( i,j ) ∈ E w ij ω ij , ∀ ω ∈ Ω 1 ( Γ ) . These constructions can be encoded in to a de Rham cochain complex 0 − − → ← − − Ω 0 ( Γ ) d − − → ← − − δ Ω 1 ( Γ ) − − → ← − − 0 , The Geometry of Synchronization Problems and Learning Group Actions 39 whic h realizes L rw 0 , the gr aph r andom walk L aplacian , as the Ho dge Laplacian of degree zero:  ∆ (0) f  i := ( δ d f ) i = 1 d i X j :( i,j ) ∈ E w ij ( f i − f j ) = ( L rw 0 f ) i , ∀ i ∈ V , ∀ f ∈ Ω 0 ( Γ ) . It is w ell known that L rw 0 diﬀers from the normalize d gr aph L aplacian L 0 b y a similarity transform L 0 = D − 1 / 2 L rw 0 D 1 / 2 , where D is a diagonal matrix with weigh ted degrees of each vertex on its diagonal. Soft ware MA TLAB co de implementing SynCut for the n umerical sim ulations and application in automated geometric morphometrics is publicly av ailable at https://github.com/trgao10/GOS- SynCut . 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