Dualities in persistent (co)homology

DUALITIES IN PERSISTENT (CO)HOMOLOGY VIN DE SIL V A, DMITRIY MOR OZO V, AND MIKAEL VEJDEMO-JOHANSSON Abstract. W e consider sequences of absolute and relative homology and cohomology groups that arise naturally for a filtered cell complex. W e establish algebraic relationships b et w een their p er- sistence mo dules, and show that they contain equiv alen t information. W e explain how one can use the existing algorithm for p ersistent homology to pro cess any of the four mo dules, and relate it to a recently introduced persistent cohomology algorithm. W e presen t exp erimental evidence for the practical efficiency of the latter algorithm. 1. Introduction The sub ject of in v erse problems deals, fundamentally , with the inference of shap e. F rom some related measuremen ts — suc h as a family of particular path in tegrals — w e try to deduce geometric information. With the classical techniques in the field, with F ourier and other integral transforms, one can deduce an impressiv e amount of information. Ho w ev er, with non-linearit y , and ill-p osed, ill-conditioned situations, the classical metho ds need increasingly large amoun ts of regularization or data cleaning. T op ology offers a family of metho ds that allo w the inference of information — if not geometric, then at least topological — in to the field. In particular, the recen t developmen t of persistent homology [1], and its applications to top ological data analysis [2], demonstrate an approac h to top ological in v ariants that b ecomes applicable to high-dimensional, finite and discrete measuremen t sets. T o take an explicit example, geological sonar in vestigations emplo y in v erse problem methods to in v estigate the geometric structure of the densit y sublev el sets in subterranean domains, relating densit y v ariations to o ccurrences of oil, water or mineral p o ck ets. The kind of information sough t starts out with a qualitativ e judgement: is there a p o c k et at all; are there several or few; are they connected or not? These first questions, before the shap e can b e given an explicit geometric description, are a matter of topological prop erties, and the study of sublevel sets of functions on domains is one of the most con vincing uses of p ersistent homology . The p ersistent homology algorithm of Edelsbrunner, Letsc her, and Zomoro dian [1] is now ten y ears old. In its natural general form [3], the input is a filtered ‘space’ (top ological space, or simplicial complex, or abstract c hain complex) and the output is a collection of half-op en real in terv als kno wn as a barco de or a persistence diagram. These barcodes contain one bar for eac h top ological feature found – one bar for each homology class, representing a hole or a higher-dimensional void. These bars come with a starting p oint, indicating the fo cal lev el at whic h the feature first b ecomes visible, and an ending p oin t, indicating the focal level at which the feature v anishes again. A fundamen tal tenet, as describ ed in [2] is that the length of suc h a bar – the difference b etw een when it shows up and when it v anishes – enco des Date : Nov em ber 27, 2024. VdS has b een partially supp orted by DARP A, through grants HR0011-05-1-0007 (TDA) and HR0011-07-1-0002 (ST oMP), and holds a Digiteo Chair. DM has b een partially supp orted by DARP A grant HR0011-05-1-0007 (TDA) and by the DOE Office of Science, Adv anced Scientific Computing Research, under aw ard num ber KJ0402-KRD047, under contract num ber DE-AC02-05CH11231. MVJ has b een partially supp orted by the Office of Na v al Research, through grant N00014-08-1-0931. 1 2 the relev ance of the feature. This emphasizes the top ological features that are en v elop ed b y a dense distribution of p oin ts, and y et hav e a geometrically large void in the middle. In man y applications, all that is required is the barco de. This tells us ho w man y homological features exist at any giv en level of the filtration, and ho w many of those surviv e to an y given sub- sequen t lev el. This information is already v ery ric h, and has b een prov en to b e statistically robust [4, 5]. Sometimes more is required. The most common request is for geometric representativ es of the features: in other w ords, explicit homology cycles representing each barco de in terv al. The original algorithm provides these cycles automatically: they are essen tial to the wa y in whic h the barco de in terv als are calculated. In fact, there are at least four natural persistent ob jects that can b e deriv ed from a filtered space. They are: p ersisten t  absolute relativ e   homology cohomology  The ‘standard’ ob ject is persistent absolute homology , and most treatmen ts fo cus on this. How ev er, it has b ecome increasingly clear that the other three ob jects are imp ortan t in their own right. The transition b etw een homology and cohomology is in some sense nothing more than the duality of v ector spaces; p ersistent homology and cohomology hav e the same barcodes. Ho w ev er, homology cycles and cohomology co cycles are quite differen t, and some applications call for cocycles rather than cycles [6]. The o ccasional utility of relative rather than absolute homology is probably easier to grasp intui tiv ely; for example see [7] for an application in sensor netw orks. It is easy to ‘fake’ the calculation of relative homology using absolute homology and a cone construction, but we point out that this trick is unnecessary . Our goal in this pap er is to pro vide a streamlined approac h to calculating barcodes and (co)cycle represen tativ es for all four persistent ob jects. W e discuss this approach in terms of abstract algebra and in terms of matrix computations. W e observ e that: • absolute homology and cohomology hav e the same barco de; • relative homology and cohomology ha v e the same barco de; • the absolute barco de and the relativ e barco de can b e deduced from each other; • the cycles and bounding c hains of persistent absolute homology determine, and are deter- mined by , the cycles and b ounding c hains of p ersisten t relative homology; • likewise, for absolute and relativ e cohomology cocycles and b ounding co chains. W e discuss tw o differen t dualities. There is the standard dualit y which in terc hanges homology and cohomology . W e call this ‘p oint wise’ dualit y . More in terestingly , there is a differen t dualit y whic h makes the follo wing in terc hange: absolute homology ↔ relative cohomology absolute cohomology ↔ relative homology W e call this ‘global’ duality , and it appears only in the con text of p ersistent top ology . Global dualit y ‘commutes’ with all possible algorithms and theorems: a metho d for calculating p ersistent absolute homology will equally well calculate persistent relative cohomology , once the input data ha v e b een turned upside-down in a particular w a y . Com bining all of these equalities and dualities, it emerges that a single calculation (run t wice) suffices to calculate all four p ersistent ob jects. Actually , we describ e t w o different algorithms for that calculation: pHcol (the ‘column algorithm’) and pHro w (the ‘ro w algorithm’). Here pHcol is essen tially the classic algorithm of [1, 3]; pHro w organises the calculation quite differently . The preferred choice dep ends, in an y given situation, on whether it is easier to lo ok up ro ws or columns 3 of the b oundary matrix of the filtered space — the sp ecific represen tation of the space usually biases this c hoice. W e are rew arded by an unexp ected pa y off. If we require only the absolute barco de, it turns out that the b est c hoice is an optimised v ersion of pHrow called pCoh (the ‘cohomology algorithm’). W e giv e exp erimental evidence to this effect. Standard practice has been to use pHcol . W e therefore call on p ersistent top ology library-writers to implement pCoh , and on p ersistent topology library-users to use it. 1.1. Outline of pap er. Section 2 is devoted to the algebra underlying this w ork. In 2.1 – 2.5 w e conduct the discussion at a high lev el (homology functors are assumed giv en, black-box style), and in 2.6 – 2.7 w e go in to the necessary chain-lev el details. In 2.8 we give a brief abstract description of the t w o dualities. Section 3 is about matrix algorithms. In 3.1 – 3.4 w e in terpret the preceding algebra in terms of matrix decomposition (following [8], again blac k-box style). In 3.5 we present the t w o algorithms, pHcol and pHro w , and explain why they giv e the same output. In Section 4 we relate the ideas in this pap er to an earlier cohomology algorithm pCoh published in [6]. W e indicate wh y w e exp ect pCoh to be faster that pHcol and pHro w for computing barco des of filtered simplicial complexes, and we verify this by exp eriment. 2. Algebra W e will assume that the reader is familiar with homology theory . Our preference is to use cellular homology , because it is a little more general than simplicial homology . 2.1. Co efficients. Individual (co)homology groups are defined with co efficients in a field k , which remains fixed throughout this pap er. P ersisten t (co)homology then has the structure of a graded mo dule o v er the polynomial ring k [ t ]. Many things go wrong when w e replace the field k with a ring, in particular the ring of in tegers Z . See [3]. 2.2. Filtered complexes. W e are interested in the p ersistent top ology of filtered top ological spaces. The simplest example is a filtered cell complex, which is a sequence X of cell complexes (2.1) X : X 1 ⊂ X 2 ⊂ · · · ⊂ X n = X ∞ where X 1 is a vertex σ 1 , and thereafter eac h complex is obtained from the previous one by adding a single cell: X i = X i − 1 ∪ σ i . Here the index set is { 1 , 2 , . . . , n } . Usually we attac h real v alues a i to the indices, which must satisfy a 1 ≤ a 2 ≤ · · · ≤ a n . Example. Our running example S will b e a cellular filtration of the 2-sphere: 1 2 1 2 3 1 2 3 4 1 2 3 4 5 1 2 3 4 5 6 1 There are six cells, σ 1 , . . . , σ 6 whic h app ear at times a i = i , for i = 1 , . . . , 6. 2.3. P ersisten t homology. If w e apply a homology functor H ( − ) to a filtered complex X w e obtain a diagram: (2.2) H ( X ) : H ( X 1 ) → H ( X 2 ) → · · · → H ( X n ) 4 T ypically H ( − ) denotes the k -dimensional homology H k ( − ; k ) or the total homology H ∗ ( − ; k ). Then (2.2) is a diagram of finite-dimensional vector spaces and linear maps, also known as a p ersistence mo dule . A persistence mo dule decomposes as a direct sum of interv al mo dules [3]. These are lab elled b y ordered pairs of in tegers [ p, q ], where 1 ≤ p ≤ q ≤ n . The pair [ p, q ] indicates a feature which p ersists o v er the index set { p, . . . , q } . W e frequen tly interpret [ p, q ] as the half-op en real interv al [ a p , a q +1 ), with the conv en tion that a n +1 = ∞ . The p ersistence diagram or barco de is the multiset of ordered pairs [ p, q ] in the decomposi- tion, or alternativ ely the m ultiset of half-open in terv als [ a p , a q +1 ). Thus w e write: P ers( H ( X )) = { [ p 1 , q 1 ] , . . . , [ p m , q m ] } = { [ a p 1 , a q 1 +1 ) , . . . , [ a p m , a q m +1 ) } It is customary in applications to discard from the p ersistence diagram those interv als [ a p , a q +1 ) for which a p = a q +1 . Example. In our running example, the intermediate spaces S 1 , S 3 , S 5 are all contractible, whereas S 2 , S 4 , S 6 are homeomorphic to the 0-sphere, 1-sphere, and 2-sphere, respectively . There are four in terv als in the persistence diagram of H ∗ ( S ): P ers( H ∗ ( S )) = { [1 , 6] 0 , [2 , 2] 0 , [4 , 4] 1 , [6 , 6] 2 } = { [1 , ∞ ) 0 , [2 , 3) 0 , [4 , 5) 1 , [6 , ∞ ) 2 } The subscript k in [ p, q ] k or [ a p , a q +1 ) k indicates that the feature occurs in k -dimensional homology . 2.4. The four standard p ersistence mo dules. The standard persistent homology mo dule H ∗ ( X ) tells us how the absolute homology groups H ∗ ( X i ) relate to each other as i v aries. W e can pla y the same game with the absolute cohomology groups H ∗ ( X i ), the relativ e homology groups H ∗ ( X n , X i ), and the relativ e cohomology groups H ∗ ( X n , X i ). Here are the four sequences, lined up for compar- ison. H ∗ ( X ) : H ∗ ( X 1 ) → . . . → H ∗ ( X n − 1 ) → H ∗ ( X n ) H ∗ ( X ) : H ∗ ( X 1 ) ← . . . ← H ∗ ( X n − 1 ) ← H ∗ ( X n ) H ∗ ( X ∞ , X ) : H ∗ ( X n ) → H ∗ ( X n , X 1 ) → . . . → H ∗ ( X n , X n − 1 ) H ∗ ( X ∞ , X ) : H ∗ ( X n ) ← H ∗ ( X n , X 1 ) ← . . . ← H ∗ ( X n , X n − 1 ) The p ersistence diagram for absolute cohomology is a multiset of integer ordered pairs [ p, q ] with 1 ≤ p ≤ q ≤ n . F or relativ e homology and cohomology , the p ersistence diagrams are m ultisets of pairs [ p, q ] with 0 ≤ p ≤ q ≤ n − 1. In all cases, we interpret [ p, q ] as the half-op en interv al [ a p , a q +1 ), with the conv en tion that a 0 = −∞ and a n +1 = ∞ . Example. In our running example, we compute P ers( H ∗ ( S 6 , S )) = { [0 , 0] 0 , [2 , 2] 1 , [4 , 4] 2 , [0 , 5] 2 } = { [ −∞ , 1) 0 , [2 , 3) 1 , [4 , 5) 2 , [ −∞ , 6) 2 } . F or instance, at index 2 w e note that there is a nontrivial element of H 1 ( S 6 , S 2 ) represented by any arc connecting the t w o p oin ts of S 2 . T o b e specific, the homology class is [ σ 3 ] = [ σ 4 ]. This class v anishes in H 1 ( S 6 , S 3 ), and so it generates the in terv al [2 , 3). The reader may detect a relationship b et w een the barco des for absolute and relativ e homology . W e formalize this in the next section. 5 2.5. Barco de isomorphisms. Prop osition 2.3. F or al l k , P ers( H k ( X )) = P ers( H k ( X )) , P ers( H k ( X ∞ , X )) = P ers( H k ( X ∞ , X )) . In other wor ds, homolo gy and c ohomolo gy have identic al b ar c o des. Pr o of. The universal co efficients theorem [9, Thm 3.2] asserts that there is a natural isomorphism H k ( X ; k ) ≡ Hom( H k ( X ; k ) , k ) . In other words, cohomology and homology are dual as v ector spaces, and hence hav e the same dimension. ‘Natural’ implies that the induced maps H k ( X i ; k ) → H k ( X j ; k ) and H k ( X i ; k ) ← H k ( X j ; k ) are adjoint, and hence hav e the same rank. Because of the wa y the barco de is uniquely determined b y dimensions and ranks, it follo ws that the absolute homology and cohomology barco des are the same. This argumen t applies equally w ell to the relativ e barcodes.  Notation. W e partition each p ersistence diagram in to t wo parts, P ers = Pers 0 ∪ P ers ∞ , where Pers 0 comprises the finite in terv als [ a, b ), and Pers ∞ the infinite in terv als [ a, ∞ ) or [ −∞ , b ). Prop osition 2.4. F or al l k , P ers 0 ( H k ( X )) = P ers 0 ( H k +1 ( X ∞ , X )) , P ers ∞ ( H k ( X )) = P ers ∞ ( H k ( X ∞ , X )) , wher e the se c ond ‘e quality’ is interpr ete d as a bije ction with [ a, ∞ ) ↔ [ −∞ , a ) . Thus, p ersistent homolo gy and r elative homolo gy b ar c o des c arry the same information, with a dimension shift for the finite intervals. The pro of app ears in Section 2.6. R emark. Th us, provided w e take the dimension shifts in to accoun t, all four barco des carry exactly the same information. If we are only in terested in barco des, we can perform calculations in any one of the four basic sequences, whic hever is the most con venien t. Since the last term of H k ( X ) is the same as the first term of H k ( X ∞ , X ), namely H k ( X n ), the t wo sequences can b e concatenated into a single sequence, whic h we denote H k ( X ) → H k ( X ∞ , X ). The index set for this sequence is { 1 , 2 , . . . , n = ¯ 0 , ¯ 1 , ¯ 2 , . . . , n − 1 } , where we use barred numerals to indicate that we are in the relative homology part of the sequence. The persistence diagram for this complex will hav e in terv als of three p ossible types: • ( p, q ) where 1 ≤ p ≤ q < n , written as [ p, q + 1) or [ a p , a q +1 ) in in terv al form. • ( ¯ p, ¯ q ) where 0 < p ≤ q ≤ n − 1, written as [ ¯ p, q + 1) or [ ¯ a p , ¯ a q +1 ). • ( p, ¯ q ) where 1 ≤ p ≤ n , 0 ≤ q ≤ n − 1, written as [ p, q + 1) or [ a p , ¯ a q +1 ). Prop osition 2.5. The b ar c o de Pers ( H k ( X ) → H k ( X ∞ , X )) c omprises the fol lowing c ol le ction of intervals: • An interval [ a, b ) for every interval [ a, b ) in P ers 0 ( H k ( X )) . • An interval [¯ a, ¯ b ) for every interval [ a, b ) in P ers 0 ( H k − 1 ( X )) . • An interval [ a, ¯ a ) for every interval [ a, ∞ ) in P ers ∞ ( H k ( X )) . 6 Pr o of. Note that the first t wo classes of in terv al in P ers ( H k ( X ) → H k ( X ∞ , X )) are those which do not meet the middle term H k ( X n ), and thus corresp ond exactly to finite interv als in P ers( H k ( X )) and Pers( H k ( X ∞ , X )). This explains the first tw o cases, once we mak e the translation P ers 0 ( H k ( X ∞ , X )) = P ers 0 ( H k − 1 ( X )). It remains to show is that the in terv als of type [ a, ¯ b ) are alwa ys of the form [ a, ¯ a ). 1 T o do this, w e need to compare the right filtration of the sequence H k ( X ) with the left filtration of the sequence H k ( X ∞ , X ). The first filtration is the nested sequence of subspaces Im( H k ( X i ) → H k ( X n )) , i = 1 , 2 , . . . , n − 1 , of H k ( X n ), and the second filtration is the nested sequence of subspaces Ker( H k ( X n ) → H k ( X n , X i )) i = 1 , 2 , . . . , n − 1 , of H k ( X n ). But the image and kernel subspaces are equal for eac h i , b y the homology long exact sequence for the pair ( X n , X i ). Thus the filtrations are the same.  R emark. The sequence H k ( X ) → H k ( X ∞ , X ) is not the same as the extended persistence [10]. The latter, defined for the sublevel sets of a real-v alued function, requires the reversal of the cells in the relativ e half of the sequence — it translates in to the use of the sup erlevel sets of the function. The meaning of extended p ersistence for a general filtered space is a lot less straight-forw ard. The most significan t difference b etw een the tw o sequences (b esides the definition) are the extended pairs, the in terv als corresponding to [ a, ∞ ) in Pers ∞ ( H k ( X )). In Prop osition 2.5 they b ecome the trivial in terv als [ a, ¯ a ); on the other hand, they are the main reason extended p ersistence was introduced: these pairs carry the new information. Another notable difference is that the dualities in this paper apply to general filtered spaces; Poincar ´ e and Lefschetz dualities inv olved in the analysis of the extended p ersistence require the domains to b e manifolds. 2.6. P ersistent c hain complexes. W e now giv e a more explicit description of the standard p er- sistence mo dules, in terms of c hain complexes. Among other things, this will lead to a clean pro of of Prop osition 2.4. Given a filtered cell complex X = σ 1 ∪ · · · ∪ σ n , define a p ersistence mo dule C : C 1 → C 2 → · · · → C n where C i = h σ 1 , . . . , σ i i , the v ector space ov er k with basis elemen ts lab elled σ 1 , . . . , σ i . W e also ha ve a b oundary map: the b oundary of any σ j is a linear combination of cells which appear previously: ∂ σ j = X i i in a decreasing order dictated by the low est non-zero en try of the column. Since the order and the columns are identical, so is the result.  R emark. Recen tly Milosavljevic, Morozov, and Skraba [11] sho wed that one can compute p ersistence in matrix m ultiplication time. R emark. One can apply the tw o algorithms of this section to the restricted matrix D p that gives only the b oundaries of the p -dimensional cells. W e can still extract some information from the R p = D p V p decomp osition of this matrix: the finite interv als [ g , h ) in dimension p − 1 and the births in dimension p , i.e. the endp oints g or h of an y p -dimensional interv al. 4. Optimiza tions 4.1. Cohomology algorithm. One of our goals has been to relate our presen t w ork to an algo- rithm pCoh for p ersistent absolute cohomology that w e described in [6]. W e based that algorithm on the idea of maintaining a right filtr ation (defined in [12]); as a result it lo oks different from pHcol and pHro w ab o ve. In fact, we now sho w that one can view pCoh as an optimization of pHro w applied to the matrix D ⊥ . W e b egin by reviewing the algorithm: 14 Algorithm 3 Cohomology algorithm pCoh . Z ⊥ = [] , birth = [] for i = 1 to n do indices = [ j | σ ∗ i ∈ δ z ∗ j , z ∗ j unmark ed in Z ⊥ ] if indices are empt y then prep end σ ∗ i to Z ⊥ and i to birth else prep end a marked σ ∗ i to Z ⊥ and i to birth p = indices [0] for j = 1 to size( indices ) do c = ( δ Z ⊥ [ indices [ j ]])[ i ] / ( δ Z ⊥ [ p ])[ i ] Z ⊥ [ indices [ j ]] = Z ⊥ [ indices [ j ]] − cZ ⊥ [ p ] mark Z ⊥ [ p ] and output the pair [ birth [ p ] , i ) = R ⊥ D ⊥ V ⊥ 0 Figure 2. The structure of matrices R ⊥ = D ⊥ V ⊥ during the execution of the row algorithm. List Z ⊥ main tains the cocycle basis for H ∗ ( X i ) in the righ t filtration order dictated b y the filtration of the space. The marking ab ov e is for exp osition only , in practice w e drop a cocycle from the list Z ⊥ as so on as it dies. When a new cell σ i en ters, it is necessarily a cocycle (since it has no cofaces), but it ma y fall into a cob oundary of a former co cycle, in whic h case (the else clause) w e up date the righ t filtration and drop the co cycle that σ i kills. T o see that this algorithm is a v ariation of the row algorithm from the previous section, observe that the co cycles that it maintains are stored in the b ottom-right corner of matrix V ⊥ during the execution of the row algorithm. Claim 4.1. The matrix Z ⊥ in the c ohomolo gy algorithm after iter ation i is e qual to the b ottom-right c orner of the matrix V ⊥ [( n − i ) ..n, ( n − i ) ..n ] after the i -th iter ation of the r ow algorithm. Pr o of. W e prov e the claim inductiv ely . Denoting with R ⊥ i , V ⊥ i , Z ⊥ i the v arious matrices after i iterations of b oth algorithms, assume the unmark ed co cycles z ∗ j in Z ⊥ i are exactly the co cycles with lo w R ⊥ ( j ) > i . In other words, the corresp onding columns R ⊥ i [ j ] = 0. F urthermore assume that the t wo matrices are iden tical, i.e. V ⊥ i = Z ⊥ i . The claim is true when i = 0. Our goal is to sho w it is true for i = k assuming it is true for i = k − 1. A t the k -th iteration, if cell σ ∗ k do es not app ear in the cob oundary of an y co cycle, then its row in R ⊥ k − 1 = δ V ⊥ k − 1 = δ Z ⊥ k − 1 is zero. It follows that it is not in the image of the map low R ⊥ k − 1 and therefore neither algorithm p erforms an y c hanges, so V ⊥ k = Z ⊥ k , and unmark ed co cycles remain as claimed. If cell σ ∗ k is in the cob oundary of a co cycle z ∗ j then k is in the image im lo w R ⊥ k − 1 . Moreo ver, from the inductiv e h yp othesis the indices j of the columns of R ⊥ k − 1 that ha ve lo w R ⊥ k − 1 ( j ) = i are exactly 15 the unmarked co cycles in Z ⊥ k − 1 that hav e σ ∗ k in their cob oundary . Therefore, the up date p erformed b y b oth algorithms is iden tical.  R emark. Since the matrix R contains the final p ersistence pairing, expressed as the map low R , the algorithm pHcol is commonly optimized to keep track only of this matrix (and ignore matrix V ). In contrast, pCoh main tains only matrix Z ⊥ = V ⊥ . 4.2. Practice. The algorithm pCoh ab ov e highlights the difference b et ween the column and the ro w versions of the p ersistence algorithm. pHcol stores all the dead cycles since it has no c hoice: an y of them migh t b e required at some future point in the reduction. pHro w , on the other hand, is able to ‘examine the future’ b y inspecting any c hosen row. It is therefore free to drop a column once it has determined its pairing and used it in the up date. pCoh do es so explicitly . In practice, such ro w access may b e difficult when computing homology: it requires quick ac- cess to the cob oundary of a given cell (since that is what a row of D is). In simplicial complex implemen tations it is common to represen t simplices as lists of vertices; then their b oundary maps are easy to compute on the fly , while their cob oundaries require a full preprocessing of the entire b oundary matrix. By switching to cohomology we turn the tables: all the primitives necessary for the ro w algorithm (and in particular the optimized v ersion giv en in this section) are readily a v ailable. 4.3. Exp eriments. The practical improv emen t resulting from these observ ations is startling. In the following table w e compare the traditional p ersistent homology algorithm pHcol with the coho- mology algorithm pCoh . W e list the total num b er of op erations performed (in terms of primitive op erations during chain arithmetic), total running time, and p eak space usage in terms of the n umber of elements stored. Dataset Algorithm Op erations Time P eak elemen ts M-50 pCoh 2,171,909,275 106 s 575,758 pHcol 609,477,028,616 4160 s 6,461,866 T-10,000 pCoh 55,930,317 6 s 22,629 pHcol 29,760,159,689 207 s 693,031 W e used the C ++ library Dion ysus [13] to p erform the ab ov e exp eriments. The homology algo- rithm pHcol in the abov e table computes only the matrix R since it suffices to extract the barco de. It also uses the original optimization of [1] and stores the non-zero co efficien ts only in the rows that corresp ond to the p ositive cells. M-50 is a filtration of an 8-skeleton of a Rips complex built on 50 random points of a Mumford dataset [14, 15] up to the maxim um pairwise distance of 1.5; the largest complex consists of 663,901 simplices. T-10,000 is an alpha shape filtration of 10,000 points sampled on a torus em b edded in R 3 ; the size of the Delaunay triangulation is 557,727 simplices. The speed-up is encouraging. W e w ould like to p oint out that these examples are not c herry-pic ked: w e ha ve yet to find a filtration on which pHcol is the faster of the tw o. Conclusion. When combined, the algebraic and experimental observ ations suggest that if giv en a c hoice, one is better off using the cohomology algorithm. Most of the time one has suc h a c hoice: for example, when computing only the p ersistence diagram. References [1] Herb ert Edelsbrunner, David Letscher, and Afra Zomorodian. T op ological persistence and simplification. Discr ete and Computational Ge ometry , 28(4):511–533, 2002. [2] Gunnar Carlsson. T op ology and data. Americ an Mathematic al So ciety , 46(2):255–308, 2009. 16 [3] Afra Zomoro dian and Gunnar Carlsson. Computing persistent homology . Discr ete and Computational Geometry , 33(2):249–274, 2005. [4] David Cohen-Steiner, Herb ert Edelsbrunner, and John Harer. Stability of p ersistence diagrams. Discr ete and Computational Ge ometry , 37(1):103–120, 2007. [5] F r´ ed´ eric Chazal, David Cohen-Steiner, Marc Glisse, Leo Guibas, and Stev e Oudot. Pro ximity of p ersistence mo dules and their diagrams. In Pr o c e e dings of the Annual Symp osium on Computational Ge ometry , pages 237– 246, 2009. [6] Vin de Silv a, Dmitriy Morozov, and Mik ael V ejdemo-Johansson. Persisten t cohomology and circular co ordinates. Discr ete and Computational Ge ometry , 45(4):737–759, 2011. [7] Vin de Silv a and Rob ert Ghrist. Co ordinate-free co verage in sensor netw orks with controlled b oundaries via homology . International Journal of R obotics Rese ar ch , 25(12):1205–1222, December 2006. [8] David Cohen-Steiner, Herb ert Edelsbrunner, and Dmitriy Morozo v. Vines and viney ards b y up dating persistence in linear time. In Pro c e e dings of the Annual Symp osium on Computational Ge ometry , pages 119–126, 2006. [9] Allen Hatcher. Algebr aic top olo gy . Cambridge Universit y Press, 2002. [10] David Cohen-Steiner, Herb ert Edelsbrunner, and John Harer. Extending persistence using poincar´ e and lefsc hetz dualit y . F oundations of Computational Mathematics , 9(1):79–103, 2009. [11] Nikola Milosa vljevic, Dmitriy Morozov, and Primoz Skraba. Zigzag p ersistent homology in matrix m ultiplication time. In Pro c e e dings of the Annual Symp osium on Computational Ge ometry , pages 216–225, 2011. [12] Gunnar Carlsson and Vin de Silv a. Zigzag p ersistence. F oundations of Computational Mathematics , 10(4):367– 405, 2010. [13] Dmitriy Morozov. Dion ysus library for computing p ersistent homology . http://www.mrzv.org/software/ dionysus . [14] Gunnar Carlsson, Tigran Ishkhanov, Vin de Silv a, and Afra Zomoro dian. On the local b ehavior of spaces of natural images. International Journal of Computer Vision , 76(1):1–12, January 2008. [15] Ann B. Lee, Kim S. Pedersen, and David Mumford. The nonlinear statistics of high-contrast patches in natural images. T ec hnical Rep ort APPTS #01-3, Division of Applied Mathematics Brown Universit y , December 2001.