math.OC 2026-03-26 0

Particle method for a nonlinear multimarginal optimal transport problem

We study a nonlinear multimarginal optimal transport problem arising in risk management, where the objective is to maximize a spectral risk measure of the pushforward of a coupling by a cost function. Although this problem is inherently nonlinear, it…

Authors: Adrien Cances, Quentin Mérigot, Luca Nenna

Particle method for a nonlinear multimarginal optimal transport problem
P AR TICLE METHOD F OR A NONLINEAR MUL TIMAR GINAL OPTIMAL TRANSPOR T PR OBLEM ADRIEN CANCES, QUENTIN M ´ ERIGOT, AND LUCA NENNA Abstract. W e study a nonlinear m ultimarginal optimal transp ort problem arising in risk managemen t, where the ob jectiv e is to maximize a spectral risk measure of the pushforw ard of a coupling by a cost function. Although this problem is inheren tly nonlinear, it is kno wn to ha ve an equiv alent linear reformulation as a multimarginal transp ort problem with an additional marginal. W e in tro duce a Lagrangian particle discretization of this problem, in whic h admissible couplings are approximated by uniformly w eighted p oin t clouds, and marginal constrain ts are enforced through W asserstein p enalization. W e prov e quan titative conv ergence results for this discretization as the num b er of particles tends to infinit y . The conv ergence rate is shown to b e go verned by the uniform quantization error of an optimal solution, and can b e b ounded in terms of the geometric prop erties of its supp ort, notably its box dimension. In the case of univ ariate marginals and supermo dular cost functions, where optimal couplings are known to b e comonotone, w e obtain sharp er con vergence rates expressed in terms of the asymptotic quantization errors of the marginals themselv es. W e also discuss the particular case of conditional v alue at risk, for which the problem reduces to a m ultimarginal partial transp ort form ulation. Finally , w e illustrate our approac h with numerical exp erimen ts in several application domains, including risk managemen t and partial barycenters, as well as some artificial examples with a repulsive cost. 1. Introduction Man y problems in risk management require estimating a sp ecific join t la w of several kno wn individual distributions. T ypically , w e hav e a function c : X 1 × · · · × X D → R whic h tak es as argument D differen t risk factors, all of which are mo deled as random v ariables X j with v alues in X j , and whose output is a scalar v alue represen ting some level of danger. F or instance, X 1 , . . . , X D could b e the c haracteristics of a river (length, width, maximal annual flo w rate, etc.) as well as some given design parameters (height of the dyke that surrounds the watercourse), and c some function giving the heigh t of the riv er in terms of the p ossible v alues for these inp ut v ariables. The randomness of the latter stems b oth from inaccuracies in their measuremen ts and from their v ariabilit y in time and/or space. W e refer to the article [7] of Io oss and Lema ˆ ıtre, which is mainly concerned with the topic of sensitivity analysis, for the details of this particular setting. In the latter, the input v ariables are assumed to b e indep enden t, but finding out the riskiest dep endence structure is an interesting problem in itself, b oth mathematically sp eaking and in terms of applications to risk management. In Io oss and Lema ˆ ıtre’s example, several industrial facilities are lo cated near the riv er, along which an artificial dyke has b een built as a precaution. But alas, there is still a risk of flo oding. Giv en some wa y to quantify the danger of any generic join t law of X 1 , . . . , X D , one ma y therefore ma y therefore b e in terested in the w orst-case scenario, in order to get a deeper understanding of the differen t risk factors at stak e. In [6], the authors study the problem in question when the risk is quantified by a generic sp e ctr al risk me asur e . 1.1. Sp ectral risk measures. W e define the spectral risk measure as a real-v alued f unc- tional on P ( R ) of the form R α ( µ ) = ˆ 1 0 F − 1 µ ( t ) α ( t ) dt, (1.1) Date : March 27, 2026. 1 2 ADRIEN CANCES, QUENTIN M ´ ERIGOT, AND LUCA NENNA with α : (0 , 1) → [0 , + ∞ ) a bounded, nondecreasing, nonnegativ e function of in tegral one, and F − 1 µ the quan tile function (or in v erse cdf ) of µ . Sp ectral risk measures are a sp ecific form of w eigh ted a v erages that are biased to wards high v alues. In particular, for an y non-constan t sp ectral function α , the functional R α is nonlinear. Note that for α ≡ 1, w e recov er the exp ected v alue E µ = ´ R z dµ ( z ), since any univ ariate probability measure is the pushforward of the uniform measure on (0 , 1) by its quantile function. Another well-kno wn risk measure is the c onditional value at risk CV aR m at lev el m ∈ (0 , 1), obtained via α ∝ 1 (1 − m, 1) . The CV aR, also kno wn as exp e cte d shortfal l or sup er quantile , is the exp ected v alue of the (normalized) restriction of µ to its fraction of mass m with highest v alues, or in other words, to its right-hand side tail of mass m , as illustrated in Figure 1. W e refer to [14] for a more comprehensiv e but concise introduction to the notion of sp ectral risk measures and their motiv ating connections with risk. CV aR m ( µ ) densit y of µ mass m R Figure 1. Conditional v alue at risk at lev el m , for an absolutely contin uous probabilit y measure µ . 1.2. The risk estimation problem. Denoting ρ j ∈ P ( X j ) the resp ectiv e marginals la ws of X 1 , . . . , X D , the problem studied in [6] reads max γ ∈ Γ( ρ 1 ,...,ρ D ) R α ( c # γ ) , ( P ) where Γ( ρ 1 , . . . , ρ D ) is the set of probability measures on the pro duct space X := X 1 · · · × . . . X D whose pro jections on the X j are the corresponding ρ j . Under very general assump- tions, the existence of a maximizer is guaran teed b y a standard compactness argumen t. Note that due to the generic sp ectral risk measure, the maximized functional γ 7→ R α ( c # γ ) is a priori nonlinear. This is a ma jor difference with resp ect to standard m ultimarginal optimal transp ort, for which the ob jective function γ 7→ ´ c dγ is linear, and corresp onds to the trivial sp ectral function α = 1. F ormulation as a line ar multimar ginal tr ansp ort pr oblem. As sho wn in [6], it turns out that ( P ) is equiv alent to the following line ar m ultimarginal optimal transp ort problem max η ∈ Γ( ρ 0 ,ρ 1 ,...,ρ D ) ˆ R × X ω c ( x ) dη ( ω , x ) , ( L ) where the additional marginal is the prob ability measure on R defined by ρ 0 := α # L (0 , 1) , and the v ariable x = ( x 1 , . . . , x D ) denotes a p oin t in the pro duct space X . The equiv alence essen tially stems from the v ariational formulation of sp ectral risk me asures, which reads R α ( µ ) = max τ ∈ Γ( ρ 0 ,µ ) ˆ R × R ω z dτ ( ω , z ) . Indeed, gluing γ and τ along the intermediate marginal µ = c # γ allows to maximize a linear ob jective o v er a single probabilit y measure η ∈ Γ( ρ 0 , ρ 1 , . . . , ρ D ), which con tains both the coupling of the D main marginals and the distribution of the w eight α o v er this coupling (thanks to the additional marginal ρ 0 ). As highlighted in [6], the ab ov e form ulation allows existing results on linear multimarginal transp ort to b e used in order to b etter understand the structure of the solutions to ( P ). P AR TICLE METHOD FOR A NONLINEAR MUL TIMARGINAL OT PROBLEM 3 1.3. Discretized problem. The solutions of optimal transp ort problems are typically sup- p orted on sets of dimension muc h smaller than that of the ambien t space — esp ecially in m ultimarginal transp ort. This renders Eulerian grid-based metho ds inherently inefficien t. In this pap er, we opt for a simple Lagrangian particle discretization, where the admissible discrete measures are those of the form δ Y := 1 N P N i =1 δ y i , with Y = ( y 1 , . . . , y N ) ∈ X N a cloud of N p oints in the pro duct space X whose positions are to b e optimized, and N a fixed p ositiv e in teger. The marginal constraints are relaxed through quadratic W asserstein p enalization terms, eac h of whic h corresp onds to the squared W asserstein distance from some prescrib ed measure ρ j to the pro jection of δ X on X j . The discre tized v ersion of ( P ) reads as the following finite-dimensional problem, which we prefer to formulate as a minimization problem, min Y ∈ X N −R α ( c # δ Y ) + λ N D X j =1 W p ( ρ j , π j # δ Y ) p . ( P N ) Here, π j : X → X j denotes the pro jection on the j -th factor of the pro duct space X = X 1 × · · · × X D , and λ N > 0 is a p enalt y parameter, whose adequate v alue will b e discussed when dealing with the conv ergence of ( P ) to ( P N ) as N → ∞ . Note that R α ( c # δ Y ) is straigh tforw ard to compute numerically . Indeed, it is b y definition a weigh ted sum of the c ( y i ), where the weigh ts dep end on the order of these v alues. Denoting p i = ´ ( i − 1 N , i N ) α ( t ) dt , w e hav e R α ( c # δ Y ) = N X i =1 p σ Y ( i ) c ( y i ) , where σ Y ∈ S N is an y p erm utation such that c ( y σ − 1 Y (1) ) ≤ · · · ≤ c ( y σ − 1 Y ( N ) ). 1.4. Con v ergence results. The conv ergence of ( P N ) to the original problem is closely link ed to the uniform quan tization of a solution of ( P ). Our tw o conv ergence results are the follo wing, and resp ectiv ely corresp ond to Corollary 17 and Corollary 24 of Section 3. W e denote τ p,d ( N ) = ( N − 1 max( p,d ) if d  = p, (log N ) 1 d N − 1 d if d = p. Theorem 1. Supp ose that the mar ginal distributions ρ j ar e c omp actly supp orte d and that c is β -H¨ older c ontinuous. L et d b e the b ox dimension (se e Definition 14) of the supp ort of some minimizer of F . Then, for the se quenc e of p enalty c o efficients λ N = τ p,n ( N ) − ( p − β ) , we have | min F N − min F | = O        N − β d if d > p, (log N ) β d N − β d if d = p, N − β p if d < p. (1.2) Mor e over, any we ak limit p oint of an arbitr ary se quenc e of minimizers δ Y N ∈ argmin F N is a minimizer of F . W e can b e more precise when the marginals are univ ariate and the cost function c is sup ermodular (see Definition 19). Theorem 2. Supp ose that the mar ginal distributions ρ j ar e univariate, i.e. X j ⊆ R , and that c is β -H¨ older and sup ermo dular. F or a pr ob ability me asur e ν on a Euclide an sp ac e, denote e p,N ( ν ) = min z 1 ,...,z N W p ν, N − 1 N X i =1 δ z i ! (1.3) 4 ADRIEN CANCES, QUENTIN M ´ ERIGOT, AND LUCA NENNA its N -p oint uniform quantization err or. Then, for the se quenc e of p enalty c o efficients λ N = h − ( p − β ) N , wher e h N = max j ∈{ 1 ,...,D } e p,N ( ρ j ) , (1.4) we have | min F N − min F | = O ( h β N ) . (1.5) Mor e over, any we ak limit p oint of an arbitr ary se quenc e of minimizers δ Y N ∈ argmin F N is a minimizer of F . Remark 3. In The or em 2, if the ρ j ar e mor e over assume d to have c onne cte d supp ort and to b e absolutely c ontinuous with density isolate d fr om 0 and ∞ on their r esp e ctive supp orts, then h N ≲ N − 1 and so the infimum of F N c onver ges to that of F at the r ate N − β . Contributions. In this pap er, w e present a particle discretization metho d for a nonlinear m ultimarginal transp ort problem studied in [6], where the maximized quantit y is some giv en sp ectral risk measure of the pushforward of the plan by a cost function c . W e describ e the Lagrangian particle discretization in question, and prov e a general quantitativ e conv ergence rate of the discretized Problem ( P N ) to the original Problem ( P ) as the num ber of particles go es to infinit y . The con v ergence rate in question corresponds to that of the optimal uniform quan tization error of some arbitrary solution of ( P ), as the num ber of Dirac masses go es to infinity . Using results by M´ erigot and Mireb eau in [10] on the uniform quantization of measures, we find an upp er b ound for the conv ergence rate in terms of the b o x dimension of the supp ort of a solution of ( P ). In [6], the authors prov e that under sup ermodularity and monotonicity assumptions on the cost function c , the comonotone plan solv es ( P ). W e lev erage this result to provide a more explicit upper b ound on the conv ergence rate, in terms of the asymptotic optimal uniform quantization errors of the marginals ρ j , under these more restrictiv e assumptions (Theorem 1). W e then deal with the sp ecific case where the sp ectral risk measure is the conditional v alue at risk, a case for which our discretization is of particular in terest. In this framew ork, Problem ( P ) b ecomes an instance of the multimarginal partial optimal transp ort problem. A t last, we show our n umerical simulations for a few interesting cases, in risk management, partial barycenters, and densit y functional theory . 2. Preliminaries and problem setting In this section, we first recall the definition of spectral risk measures, as w ell as their v ariational formulation. W e then give the corresp onding reformulation of Problem ( P ) in to an ordinary m ultimarginal transp ort Problem ( L ) with an additional marginal induced by the risk measure, an equiv alence first established in [6]. Assumptions. Throughout the whole pap er, the X j ⊆ R d j are conv ex subsets of Euclidean spaces (with p otentially different dimensions d j ) and the ρ j ∈ P p ( X j ) are probability mea- sures with finite p -moment, where p ∈ [1 , ∞ ) is a fixed finite exp onent. As a direct conse- quence, the pro duct set X = X 1 × · · · × X D is also conv ex — we naturally endow it with Euclidean norm of the em bedding pro duct space R d 1 × · · · × R d D — and w e ha v e the inclusion Γ( ρ 1 , . . . , ρ D ) ⊆ P p ( X ), i.e. an y joint measure of the ρ j also has finite p -moment. F or now, w e only assume the cost function c : X → R to b e measurable. In the introduction, we assumed that the sp ectral function α was b ounded. In fact, up to restricting the domain of R α to a smaller set of probabilit y measure, it suffices to assume α is in some Leb esgue space L q (0 , 1), q ∈ (1 , ∞ ]. In the remainder of this article, we adopt this more general framew ork, whic h accommo dates sp ectral risk measures that exhibit an ev en stronger preference for large v alues. Definition 4. A sp ectral risk measure is a functional R α : P ¯ q ( R ) → R of the form R α ( µ ) = ˆ 1 0 F − 1 µ ( t ) α ( t ) dt, (2.1) P AR TICLE METHOD FOR A NONLINEAR MUL TIMARGINAL OT PROBLEM 5 wher e α : (0 , 1) → [0 , + ∞ ) is a nonne gative, nonde cr e asing function of inte gr al one in L q (0 , 1) for some q ∈ (1 , ∞ ] , c al le d sp ectral function . H¨ older’s ine quality ensur es that R α is inde e d wel l-define d on P ¯ q ( R ) , wher e ¯ q is the c onjugate H¨ older exp onent of q . Remark 5 (Domain of the sp ectral risk measure) . If α ∈ L q (0 , 1) for some q ∈ (1 , ∞ ] , the right-hand side of (2.1) is wel l-define d and finite for a much lar ger c ol le ction of pr ob ability me asur es than P ¯ q ( R ) . F or instanc e, sinc e α is b oth nonne gative and nonde cr e asing, it suffic es that µ has left-tail with finite first moment and right-tail with finite ¯ q -moment for the inte gr al in (2.1) to b e wel l-define d. Mor e over, de c omp osing quantile functions into ne gative and p ositive p arts and using F − 1 µ s ≤ max { F − 1 µ 0 , F − 1 µ 1 } for µ s = (1 − s ) µ 0 + sµ 1 , it is str aightforwar d to show that the domain of any sp e ctr al risk me asur e is a c onvex subset of P ( R ) . W e will sometimes refer to R α as the α -risk , as α uniquely determines the sp ectral risk measure. Note that b y monotonicit y of α , its set of discon tin uit y points is at most coun table, and th us Leb esgue negligible, which means R α is unaffected by the v alues at these p oin ts. Remark 6. F or α ≡ 1 we natur al ly r etrieve the exp e cte d value, sinc e ( F − 1 µ ) # L (0 , 1) = µ . On the other hand, taking α m = m − 1 1 (1 − m, 1) yields the c onditional value at risk (or exp e cte d shortfal l) at level m ∈ (0 , 1] , which we denote CV ar m ( µ ) . The latter is define d as the exp e cte d value of the (normalize d) r estriction of µ to its fr action of mass m with highest values. Sp ectral risk measures ha v e the follo wing w ell kno wn v ariational characterization in terms of one-dimensional optimal transp ort with the standard bilinear cost. Lemma 7. L et R α b e a sp e ctr al risk me asur e and denote q ∈ (1 , ∞ ] an exp onent such that α ∈ L q (0 , 1) . Then, for any µ ∈ P ¯ q ( R ) we have R α ( µ ) = max τ ∈ Γ( α # L (0 , 1) ,µ ) ˆ R × R z ω dτ ( ω , z ) . (2.2) In p articular, R α is c onc ave on P ¯ q ( R ) . Equivalenc e of ( P ) with a line ar pr oblem. Before addressing Lagrangian discretization, let us briefly precise the equiv alence of Problem ( P ) with a linear transp ort problem that has an additional marginal. This equiv alence was first highligh ted by Enna ji et al. (see [6, Theorem 18]) and is a direct consequence of the v ariational form ulation describ ed in Lemma 7 for sp ectral risk measures. Prop osition 8 ([6, Theorem 18]) . Consider the line ar multimar ginal pr oblem max η ∈ Γ( ρ 0 ,ρ 1 ,...,ρ D ) ˆ R × X s dη , ( L ) wher e ρ 0 = α # L (0 , 1) ∈ P ( R ) and s ( ω , x ) := ω c ( x ) for ( ω , x ) ∈ R × X . A pr ob ability me asur e η is a solution to ( L ) if and only if the fol lowing c onditions hold: • the pr ob ability me asur e γ := π X # η ∈ Γ( ρ 1 , . . . , ρ D ) is a solution to Pr oblem ( P ) , • the pushforwar d τ := ( π 0 , c ◦ π X ) # η ∈ Γ( ρ 0 , c # γ ) has monotone incr e asing supp ort, wher e π X ( ω , x ) := x j ∈ X j and π 0 ( ω , x ) := ω ∈ R ar e the c anonic al pr oje ctions for the pr o duct sp ac e R × X = R × X 1 × · · · × X D . 3. Discretiza tion and convergence resul ts W e wish to n umerically solve the following c onvex optimization problem max γ ∈ Γ( ρ 1 ,...,ρ D ) R α ( c # γ ) . ( P ) The Lagrangian particle discretization scheme we inv estigate reads min Y ∈ X N −R α ( c # δ Y ) + λ N D X j =1 W p ( ρ j , π j # δ Y ) p , ( P N ) 6 ADRIEN CANCES, QUENTIN M ´ ERIGOT, AND LUCA NENNA with λ N > 0 a p enalty parameter whose dep endence on the num ber of particles N w e shall deal with in the presen t section. Note that thanks to the ρ j ha ving finite p -momen t, the p enalt y terms are all finite. In order to prov e (some notion of ) conv ergence of ( P N ) to ( P ), one needs some control on the sp ectral risk measures of the c -pushforwards in terms of the joint measures themselves. T o this end, w e make the following assumptions. Assumption 1. We have p ∈ [1 , ∞ ) , and ther e exists q ∈ (1 , ∞ ) and β ∈ (0 , 1] such that (1) α ∈ L q (0 , 1) , (2) c is β -H¨ older, with H¨ older c onstant C H , (3) β p + 1 q ≤ 1 . We denote ¯ q ∈ (1 , ∞ ) the c onjugate H¨ older exp onent of q , i.e. 1 /q + 1 / ¯ q = 1 . Remark 9. Under these assumptions, Pr op osition 8 stil l holds without any b ounde dness assumption on α . Unless stated otherwise, the symbol ≲ will hide p ositiv e constan ts that only dep end on D , p , q , α , β , and c . The follo wing lemma, v alid under the ab o v e assumptions, is a k ey ingredien t for the conv ergence results that will follo w. It is merely an extension of the Lipsc hitz con tin uit y results in [6, Lemma 33, Prop osition 34] to the more general case of H¨ older cost functions and L p sp ectral functions. Lemma 10 (H¨ older con tinuit y of the ob jective function) . Supp ose that Assumption 1 holds. Then, R α ( c # · ) is finite on the c onvex set P p ( X ) , and for any γ , ˜ γ ∈ P p ( X ) we have |R α ( c # γ ) − R α ( c # ˜ γ ) | ≤ ∥ α ∥ L q C H W p ( γ , ˜ γ ) β . (3.1) In other wor ds, the map γ 7→ R α ( c # γ ) is b oth finite and β -H¨ older on the p -Wasserstein sp ac e on X , with H¨ older c onstant ∥ α ∥ L q C H . Pr o of. W e first show that c pushes forwards an y measure with finite p -moment to a measure with finite ¯ q -moment. Indeed, for γ ∈ P p ( X ), the ¯ q -momen t of c # γ with respect to any giv en x o ∈ X writes ˆ X | c ( x ) − c ( x o ) | ¯ q dγ ( x ) ≤ C H ˆ X | x − x o | β ¯ q dγ ( x ) . The last integral is equal to W β ¯ q ( γ , δ x o ) to the p o w er β ¯ q , and since the third p oin t of Assumption 1 reads β ¯ q ≤ p , this W asserstein distance is b ounded b y the finite quantit y W p ( γ , δ x o ). This sho ws that c # γ ∈ P ¯ q ( X ), whic h implies R α ( c # γ ) is finite thanks to H¨ older’s inequalit y and to α being L q . If ˜ γ is another measure in P p ( X ), H¨ older’s inequalit y also yields |R α ( c # γ ) − R α ( c # ˜ γ ) | ≤ ˆ 1 0 α | F − 1 c # γ ∗ − F − 1 c # γ N | ≤ ∥ α ∥ L q ∥ F − 1 c # γ − F − 1 c # ˜ γ ∥ L ¯ q . Since ∥ F − 1 c # γ − F − 1 c # ˜ γ ∥ L ¯ q = W ¯ q ( c # γ , c # ˜ γ ), it remains to control the distance b et w een the t wo pushforw ards. Precisely , β -H¨ older regularit y of c implies that W ¯ q ( c # γ , c # ˜ γ ) ≤ C H W β ¯ q ( γ , ˜ γ ) β , and since by assumption β ¯ q ≤ p the desired inequality follows. □ T o study conv ergence of ( P N ) to ( P ), define the functionals F , F N : P ( X ) → R ∪ { + ∞} b y ( F ( γ ) = −R α ( c # γ ) + χ Γ( ρ 1 ,...,ρ D ) ( γ ) , F N ( γ ) = −R α ( c # γ ) + λ N P D j =1 W p ( ρ j , π j # γ ) p + χ D N ( γ ) , where D N = { δ Y : Y ∈ X N } is the domain of the discretized functional, consisting of all uniform clouds of N p oin ts in the pro duct space. Of course, Problems ( P ) and ( P N ) P AR TICLE METHOD FOR A NONLINEAR MUL TIMARGINAL OT PROBLEM 7 resp ectiv ely corresp ond to the minimization of F and F N . Here, the notation χ E corresp onds to the characteristic function of the set E in the sense of conv ex analysis: χ E ( x ) = 0 if x ∈ E , + ∞ otherwise. Prop osition 11 (Existence of solutions) . Under Assumption 2, Pr oblems ( P N ) and ( P ) b oth have at le ast one solution. Pr o of. Thanks to finite p -moment of the marginal distributions ρ j and to con tinuit y of the ob jective function with resp ect to W p , existence of a solution to ( P ) follows from standard argumen ts. T o see that ( P N ) also has a solution, consider a minimizing sequence { δ Y ℓ } ∞ ℓ =0 ⊆ D N . By co ercivit y of the p enalt y terms with resp ect to the p ositions of the particles, the latter remain in some fixed compact set throughout the sequence. Up to extraction of a subsequence, we can therefore assume that Y ℓ con v erges to some limit cloud Y as ℓ → ∞ , and uniform b oundedness of the Y ℓ ensures that the corresp onding probabilit y measures δ Y ℓ con v erge to δ Y for the W p distance. Con v ergence of the v alues F N ( δ Y ℓ ) to F N ( δ Y ) then directly follows from contin uit y of R α ( c # · ) and of pro jections on subspaces, with resp ect to the p -W asserstein distance. □ 3.1. Con v ergence of the discretized problems. W e start by analyzing the t wo differ- en t parts for the study of the conv ergence. In particular in order to get an estimation of (min F N − min F ) w e provide an upp er and a low er b ound and then optimize them in order to find the optimal λ N giving the matching b ound. W e describ e briefly the strategy of the pro of b efore detailing the rigorous step: • Upp er b ound. T o estimate (min F N − min F ), one m ust approach a fixed minimizer γ ∗ ∈ argmin F b y some N -p oint uniform cloud, and thanks to the control on the sp ectral risk measures giv en b y Lemma 10, it suffices that the p oin t cloud be close to γ ∗ for the p -W asserstein distance. The p enalt y terms will indeed also b e controlled, since pro jecting t w o measures one some common set can only decrease their W asserstein distance. In the end, we get a b ound of the form min F N − min F ≲ u β N + λ N u p N , (3.2) where u N is the p -W asserstein distance from the considered N -p oin t uniform cloud to γ ∗ . T o minimize the right-hand side, one first naturally takes the p oin t cloud which minimizes the latter distance, so that u N is the optimal N -p oin t uniform quan tization error of γ ∗ with resp ect to W p . Note that since the minimizer γ ∗ is arbitrary , prop erties of a single minimizer can b e harnessed to obtain asymptotic b ounds. • Lo w er b ound. On the other hand, low er bounding (min F N − min F ) requires to approac h a p oin t cloud δ Y ∗ N ∈ argmin F N minimizing the discretized functional b y some join t measure γ of the marginals ρ j . W e exploit the discrete nature of the p oin t cloud to construct a particular joint measure γ N whose distance to the p oin t cloud is controlled by that of the resp ectiv e pro jections. This constructed measure is a blo c k approximation made of N equal-mass blo c ks — one for each of the N points of the cloud — induced by the W p - optimal transp orts from the ρ j to the resp ectiv e pro jections of the p oin t cloud. Plugging in γ N to upp er b ound the minimum of F , we thus hav e min F N − min F ≳ λ N v p N − C v β N , (3.3) where v N denotes the p -W asserstein distance from γ N to δ Y ∗ N and C > 0 is some p ositiv e constan t dep ending on the cost function c , on the sp ectral function α , and on D the num b er of marginals. • Matc hing bound. T aking now λ N =  1 u N  p − β in the upp er b ound and using mainly Y oung’s inequality in the lo wer b ound, (3.2) and (3.3) rewrite resp ectively min F N − min F ≲ u β N , 8 ADRIEN CANCES, QUENTIN M ´ ERIGOT, AND LUCA NENNA min F N − min F ≳ − λ − β p − β N , from which the matching b ound easily follows | min F N − min F | = O ( u β N ) . W e now explicitly construct the joint measure of the ρ j used to approach a uniform cloud of N p oin ts. Lemma 12 (Blo c k approximation) . L et δ Y ∈ D N b e a uniform cloud of N p oints in X , and denote δ Y j its pr oje ction on X j . Ther e exists a joint me asur e γ ∈ Γ( ρ 1 , . . . , ρ D ) such that W p ( γ , δ Y ) ≲   D X j =1 W p ( ρ j , δ Y j ) p   1 p . (3.4) Pr o of. W e denote y 1 , . . . , y N ∈ X the p oin ts of the cloud. F or each j , denote ρ j = P N i =1 ρ i,j the uniform decomp osition of ρ j induced by its W p -optimal transp ort to δ Y j . That is, the p ositiv e measure ρ i,j of mass 1 / N is sent to y i,j ∈ X j , the j -th co ordinate of y i . W e patc h to- gether the comp onents corresponding to the same index i by defining γ = N D − 1 P N i =1 ( ρ i, 1 ⊗ · · · ⊗ ρ i,D ), where N D − 1 is a normalization constant. Then γ ∈ Γ( ρ 1 , . . . , ρ D ), and its W asserstein distance to the p oin t cloud δ Y can b e estimated by sending eac h patc h to its corresp onding Dirac mass. T o decouple the D comp onen ts in the distance from a p oint x to some Dirac p osition y i , we use equiv alence of norms in R D , which yields W p ( γ , δ Y ) p ≤ N D − 1 N X i =1 ˆ X ∥ x − y i ∥ p d ( ρ i, 1 ⊗ · · · ⊗ ρ i,D )( x ) ≲ N D − 1 N X i =1 ˆ X D X j =1 ∥ x j − y i,j ∥ p d ( ρ i, 1 ⊗ · · · ⊗ ρ i,D )( x ) = D X j =1 N X i =1 ˆ X j ∥ x j − y i,j ∥ p dρ i,j ( x j ) ! = D X j =1 W p ( ρ j , δ Y j ) p , where, in the last line, w e used the fact that sending eac h ρ i,j to y i,j is a W p -optimal transport from ρ j to δ Y j . Note that the choice of the pro duct measure for each blo c k is not imp ortan t for the pro of, as long as each blo c k has the correct marginals. □ Thanks to the considerations in the b eginning of this subsection, we hav e the following con v ergence result. Theorem 13. Supp ose that Assumption 1 holds, and let u N b e some upp er b ound on the W p -optimal uniform quantization err or for some fixe d minimizer γ ∗ of F . Then, letting λ N = u − ( p − β ) N , we have | min F N − min F | = O ( u β N ) . (3.5) Mor e over, for any se quenc e of minimizers δ Y ∗ N ∈ argmin F N , N ∈ N , every limit p oint is a minimizer of F , and for every j ∈ { 1 , . . . , D } we have W p ( ρ j , π j # δ Y ∗ N ) = O ( u N ) . (3.6) Pr o of. The estimate on (min F N − min F ) has already b een dealt with in the b eginning of this section, so we fo cus on the full pro of for the upp er b ound on (min F − min F N ). T ake P AR TICLE METHOD FOR A NONLINEAR MUL TIMARGINAL OT PROBLEM 9 δ Y ∗ N a minimizer of F N and let γ N ∈ Γ( ρ 1 , . . . , ρ D ) b e the joint measure giv en b y Lemma 12. Thanks to the latter and to β -H¨ older contin uit y of R α ( c # · ) deriv ed in Lemma 10, w e hav e min F N − min F ≳ − C W p ( γ N , δ Y ∗ N ) β + λ N W p ( γ N , δ Y ∗ N ) p where C is constant that only dep ends on p , q , α , and c . W e then apply Y oung’s inequalit y with exp onen ts p/β and its conjugate to find C W p ( γ N , δ Y ∗ N ) β =  p β λ N W p ( γ N , δ Y ∗ N ) p  β p  p β λ N C − p β  − β p ≤ λ N W p ( γ N , δ Y ∗ N ) p + p − β p  p β λ N C − p β  − β p − β , and the estimate (3.5) follows from λ B = u − ( p − β ) B . Let us no w prov e the b ound (3.6) on the marginal discrepancies. W e denote S N = P D j =1 W p ( ρ j , π j # δ Y ∗ N ) p . Thanks to S N ≤ W p ( δ Y ∗ N , γ ) p for every γ ∈ Γ( ρ 1 , . . . , ρ D ) and to β -H¨ older contin uity of R α ( c # · ), we hav e F N ( δ Y ∗ N ) ≥ (min F − C S β /p N ) + λ N S N . By optimalit y of δ Y ∗ N , Equation (3.5) writes F N ( δ Y ∗ N ) ≤ min F + C ′ u β N , and together with the previous inequality w e obtain − C S β /p N + λ N S N ≤ C ′ u β N . (3.7) W e apply Y oung’s inequality to write C S β /p N ≤ 1 2 λ N S N + C ′′ λ − β / ( p − β ) N , and injecting this inequalit y in (3.7) yields, after simplification, λ N S N ≲ u β N , whic h implies S N ≲ u p N thanks to λ N = u − ( p − β ) N . □ Uniform quantization. W e no w briefly in troduce the uniform quan tization of measures on an arbitrary metric space Y , and state a result of [10] linking the asymptotic uniform quan- tization error of a measure and the dimension of its supp ort. W e consider quantization with resp ect to the p -W asserstein distance, with p ∈ [1 , ∞ ) an arbitrary exp onen t. Let ν ∈ P p ( Y ) b e a probability measure with finite p -moment. Its N -p oint uniform quantization err or of order p is defined by e p,N ( ν ) = inf z 1 ,...,z N ∈Y W p ( ν, N − 1 P N i =1 δ z i ) , (3.8) and a discrete measure N − 1 P N i =1 δ z i for whic h the z i form a minimizer is called an optimal ( N -p oint) uniform quantizer of ν . Up to considering the completion of Y rather than Y directly , the infimum is alw ays achiev ed and ν has optimal uniform quantizers of all cardi- nals. Before discussing estimates on the asymptotic quantization error rate, let us recall the definition of b o x dimension for compact metric spaces. Definition 14. L et K b e a c omp act metric sp ac e. F or any p ositive inte ger N , define the optimal N -p oint cov ering radius by r N ( K ) = min ( r ≥ 0 : ∃ x 1 , . . . , x N ∈ K, K ⊆ N [ i =1 ¯ B ( x i , r ) ) , (3.9) 10 ADRIEN CANCES, QUENTIN M ´ ERIGOT, AND LUCA NENNA wher e the infimum is inde e d a minimum thanks to c omp actness of K and to the c overs c onsisting of close d b al ls. We c al l (upp er) b ox dimension — or Minko wski dimension — of K the nonne gative sc alar d box ( K ) = lim sup N →∞ log N − log r N ( K ) . (3.10) Remark 15. Box dimension is always an upp er b ound for Hausdorff dimension, and is much simpler to c ompute, as it do es not involve me asur es, in p articular. Both dimensions c oincide for c omp act manifolds, but differ in gener al. F or instanc e, while al l c ountable sets have Hausdorff dimension zer o, we have d box  ([0 , 1] ∩ Q ) d  = d, d box  1 n : n ∈ N ∗  = 1 2 . In [10, Prop osition 12], M ´ erigot and Mireb eau make a link b etw een the asymptotic uniform quan tization e rror of a measure and the (upp er) b o x dimension of its supp ort. They state their result in a generic metric space, and although only quantization with resp ect to the quadratic W asserstein distance is needed in their w ork, the pro of extends mutatis mutandis to the case of a general W asserstein exp onent p ∈ [1 , ∞ ). Prop osition 16 ([10, Prop osition 12]) . Consider an exp onent p ∈ [1 , ∞ ) and a c omp actly supp orte d pr ob ability me asur e ν ∈ P ( Y ) on a metric sp ac e Y . If the supp ort of ν has finite b ox dimension d = d box ( E ) , then e p,N ( ν ) ≲ τ p,d ( N ) , (3.11) wher e τ p,d ( N ) = ( N − 1 max( p,d ) if d  = p, (log N ) 1 d N − 1 d if d = p, (3.12) and wher e ≲ hides a c onstant which only dep ends on p , d , and on the set spt ν . Thanks to the ab o v e prop osition, we deduce the following conv ergence result as an imme- diate corollary of Theorem 13. Corollary 17. Supp ose that Assumption 1 holds and that the ρ j ar e al l c omp actly supp orte d. L et d ∈ [1 , D ] b e the b ox dimension of the supp ort of some minimizer of F . Then, letting λ N = τ p,d ( N ) − ( p − β ) wher e τ p,d is define d in (3.12) , we have | min F N − min F | = O        N − β d if d > p, (log N ) β d N − β d if d = p, N − β p if d < p, (3.13) wher e O hides a c onstant which dep ends on the supp ort of the minimizer in question. Mor e- over, for any se quenc e of minimizers δ Y ∗ N ∈ argmin F N , N ∈ N , every we ak limit p oint is a minimizer of F . Remark 18 (Univ ariate marginals) . In the c ase of univariate mar ginals with c omp act sup- p orts and under a few assumptions on the c ost function c , a minimizer of F wil l have supp ort of dimension d = 1 , le ading to an asymptotic c onver genc e r ate β /p . In the next se ction, we wil l se e that when the c ost function is sup ermo dular, one c an in fact obtain a much faster c onver genc e r ate, which dir e ctly dep ends on the asymptotic uniform quantization err ors of the fixe d (unidimensional) mar ginals ρ j . 3.2. The case of a supermo dular cost function. In this section, we assume that the marginals are univ ariate measures, that eac h X j ⊆ R is an in terv al, and that the cost function c : X 1 × · · · × X D → R is sup ermo dular, see Definition 19. Submo dularit y has gained ground in the multimarginal optimal transp ort literature, with a seminal result by Carlier in [3] for strictly sup ermo dular costs, and an extension to the case of multidimensional v ariables by P ass in [12]. P AR TICLE METHOD FOR A NONLINEAR MUL TIMARGINAL OT PROBLEM 11 Definition 19. A function c : I 1 × · · · × I D → R define d on some pr o duct of r e al intervals I j ⊆ R is said to b e sup ermodular if for every x, ˜ x ∈ I 1 × · · · × I D we have c ( x ∧ ˜ x ) + c ( x ∨ ˜ x ) ≥ c ( x ) + c ( ˜ x ) , (3.14) wher e x ∧ ˜ x and x ∨ ˜ x r esp e ctively denote the c o or dinate-wise minimum and maximum of x and ˜ x . We say c is strictly sup ermo dular when the e quality is strict for any p air of distinct p oints x  = ˜ x . F or C 2 functions, sup ermo dularit y is equiv alen t to ∂ 2 c ∂ x j ∂ x k ≥ 0 on the whole domain for ev ery j  = k , and the strict version of these inequalities implies strict sup ermo dularit y , although it is not a nece ssary condition. Sup ermo dularity and c omonotonicity. It is a standard result that for a strictly supermo dular cost function c , the supp ort of an y γ maximizing E[ c # γ ] ov er Γ( ρ 1 , . . . , ρ D ) is total ly ordered for the co ordinate-wise order on R D . This may b e seen as an immediate consequence of the w ell-kno wn c -cyclical monotonicit y of the supp ort of any optimal transport plan. T otal ordered-ness of the supp ort is a very strong prop ert y , as it completely determines the join t measure in terms of the prescrib ed marginals. The unique coupling with totally ordered supp ort is called the c omonotone plan, and reads γ mon := ( F − 1 ρ 1 , . . . , F − 1 ρ D ) # L (0 , 1) . The reader may consult [5] for a rather extensive bibliographic ov erview of comonotonicity and its applications in risk management. As highlighted in [6, Lemma 21], optimality of the comonotone plan for standar d multi- marginal transp ort with a sup ermo dular cost may b e generalized to the sp ectral risk mea- sures framework, up to the additional assumption of nondecreasing monotonicity of the cost in each v ariable. This additional assumption ensures that the w eigh ting α increases with the co ordinate-wise order along the comonotone supp ort of γ . W e summarize the result in question in the follo wing lemma, which still holds for L q sp ectral functions under our assumptions. Lemma 20 ([6, Lemma 21]) . In addition to Assumption 1, supp ose that c : X 1 ×· · ·× X D → R is sup ermo dular, and nonde cr e asing in e ach variable. Then the c omonotone c ouplings of ρ 1 , . . . , ρ D and of ρ 0 , ρ 1 , . . . , ρ D r esp e ctively solve Pr oblems ( P ) and ( L ) , and their c ommon maximal value is ˆ 1 0 c ( F − 1 ρ 1 ( t ) , . . . , F − 1 ρ D ( t )) α ( t ) dt. (3.15) Mor e over, if c is strictly sup ermo dular, and strictly incr e asing in e ach variable, then the solution of ( L ) is unique on the supp ort of α , in the sense that any other solution c oincide with the c omonotone plan on the set (0 , + ∞ ) × X 1 × · · · × X D . This natur al ly tr anslates to uniqueness of the solution of ( P ) when r estricte d to its fr action of mass m = 1 − inf spt α that is maximal for the c o or dinate-wise or der on R D . Remark 21 (Relaxing the supermo dularity assumption) . In the c ase of a generic sup ermo d- ular c ost function s for a line ar multimar ginal optimal tr ansp ort pr oblem, the c omonotone plan is a solution that do es not dep end on s . In the sp e ctr al risk me asur e fr amework of L emma 20, not only is the c omonotone plan indep endent of the (sup ermo dular) c ost function c , but it is also indep endent of the sp e ctr al function α , and a fortiori of the sp e ctr al risk me a- sur e c onsider e d. Sup ermo dularity is inde e d a very str ong and r estrictive assumption. Y et, we emphasize that for a lar ge class of non-sup ermo dular c ost functions, for which the solu- tion do es dep end on the given sp e ctr al risk me asur e, the said solution has a low-dimensional supp ort. In such c ases, Cor ol lary 17 gives a sharp c onver genc e r ate. We r efer to the work of Pass [12, 11] for various criteria yielding b ounds on the dimension of the supp ort, as wel l as his joint work [8] with Kim for a gener al c ondition under w hich any solution is c onc entr ate d on a gr aph of one of the mar ginal variables. 12 ADRIEN CANCES, QUENTIN M ´ ERIGOT, AND LUCA NENNA Remark 22 (The compatibilit y condition) . In [6] , the authors mention a c ondition known as compatibilit y , first intr o duc e d by Pass in [12, Sect 6] as an invariant form of the sup ermo du- larity c ondition. Inde e d, a c ost function of D variables is (strictly) c omp atible if and only if it is (strictly) sup ermo dular up to a change of sign of some subset of the D variables. It fol lows that any r esult that holds under the (strict) sup ermo dularity assumption dir e ctly extends to the c ase of (strict) c omp atibility, with the ade quate sign changes. In p articular, L emma 20 stil l holds for c omp atible c ost functions, up to r eplacing the incr e asing monotonicity assump- tion by de cr e asing monotonicity and taking the r everse function t ∈ (0 , 1) 7→ F − 1 ρ j (1 − t ) inste ad of the standar d quantile function, for the r elevant set of variables. We shal l r efer to the optimal plans in question for Pr oblems ( P ) and ( L ) as the c -monotone plan and the s -monotone plan , r esp e ctively. The pro of of our conv ergence result in the case of a sup ermo dular cost function relies on the fact that, broadly sp eaking, the comonotone coupling can b e quan tized as well as its marginals, as stated in the next prop osition. Prop osition 23 (Quan tizing the comonotone coupling) . Supp ose that the mar ginals ar e one- dimensional, i.e. X j ⊆ R for al l j , and let γ mon = ( F − 1 ρ 1 , . . . , F − 1 ρ D ) # L (0 , 1) b e the c omonotone c oupling of the ρ j . Then, for every N ∈ N 1 , we have e p,N ( γ mon ) ≲   D X j =1 e p,N ( ρ j ) p   1 p , (3.16) and the c omonotone c oupling of W p -optimal uniform quantizers of the ρ j yields a W p - quantization err or of γ mon no gr e ater than the right-hand side. Pr o of. F or eac h j , let δ Y j = N − 1 P N i =1 δ y i,j b e an optimal uniform quantizer of ρ j in the sense of W p , and assume without loss of generalit y that the p ositions of the Dirac masses are ordered, i.e. y 1 ,j ≤ · · · ≤ y N ,j . Since w e wan t to approac h the comonotone coupling of the ρ j , w e naturally pair these one-dimensional p oint clouds in a monotone wa y . That is, we define δ Y = N − 1 P N i =1 δ y i where y i = ( y i,j ) N j =1 ∈ R D . By monotonicity , the optimal transp ort from γ mon to δ Y in the sense of W p consists in sending the i -th low est part of the comonotone plan to the i -th Dirac mass of low est co ordinates. More precisely , we send the submeasure γ i = ( F − 1 ρ 1 , . . . , F − 1 ρ D ) # L ( i − 1 N , i N ) of mass 1 / N to the p oint y N i . Note that γ i is a join t measure — in fact, the unique comonotone one — of the i -th lo west submeasures in the resp ective uniform decomp ositions of eac h marginal ρ j in to N ordered parts. By equiv alence of norms in R D , we thus find W p ( γ mon , δ Y ) p = N X i =1 ˆ R D ∥ x − y i ∥ p dγ i ( x ) ≲ N X i =1 ˆ R D   D X j =1 | x j − y i,j | p   dγ i ( x ) = D X j =1 N X i =1 ˆ R D | x j − y i,j | p d [ π j # γ i ]( x j ) ! , and the sum in paren theses is the optimal cost e p,N ( ρ j ) p = W p ( ρ j , δ Y j ) p , since the monotone transp ort of γ mon to δ Y pro jects to comonotone transp orts from the ρ j to their resp ectiv e optimal quan tizers δ Y j . □ F rom Prop osition 23 w e deduce the follo wing con v ergence result, as an immediate corollary of Theorem 13. Note that w e do not need an y compactness assumption on the ρ j . P AR TICLE METHOD FOR A NONLINEAR MUL TIMARGINAL OT PROBLEM 13 Corollary 24. Supp ose that Assumption 1 holds and that the c ost function c is sup ermo d- ular, and monotone incr e asing in e ach variable. Then, letting λ N = h − ( p − β ) N wher e h N = max j ∈{ 1 ,...,D } e p,N ( ρ j ) , (3.17) we have | min F N − min F | = O ( h β N ) . (3.18) Mor e over, for any se quenc e of minimizers δ Y ∗ N ∈ argmin F N , N ∈ N , every we ak limit p oint is a minimizer of F . Uniform quantization in dimension one. Bencheikh and Jourdain ha v e recently obtained a necessary and sufficient conditions for the asymptotic uniform quan tization error of a univ ariate probability densit y ν ∈ P p ( R ) to go to zero with order θ ∈ (0 , 1 p ). Their main result [1, Theorem 2.2] reads sup N ∈ N 1 N θ e p,N ( ν ) < ∞ ⇐ ⇒ sup x ≥ 0 x p 1 − θp ( F ν ( − x ) + 1 − F ν ( x )) , (3.19) where the supremum on the right-hand side is essentially an estimate on the left and right tail distribution of ν . The authors also recall the fact that for measures with un bounded supp ort, the error cannot go to zero with order strictly greater than 1 p . On the other hand, when ν is compactly supp orted, the order of conv ergence for the uniform quan tization error is at least 1 p , and the b est p ossible order is θ = 1. A well- kno wn sufficient condition for order θ = 1 is that ν is absolutely con tinuous with supp ort a compact interv al, and densit y isolated from zero on this interv al. Broadly sp eaking, when ν has connected compact supp ort, it is the lo cal b eha vior of the density near its zeros that determines the con v ergence rate of the quantization error. In a first version of [1], Bencheikh and Jourdain prov e that for any a > 1, the probability density ρ ( x ) = ax a − 1 with supp ort [0 , 1] has a uniform quantization error going to zero at order exactly min { 1 p + 1 a , 1 } , with a logarithmic mu ltiplicative correction factor (log N ) 1 p for the limit case 1 p + 1 a = 1. Comp arison criterion. As a final remark concerning uniform quantization in dimension one, w e giv e a simple criterion for upper and low er bounds on the asymptotic error rate of a densit y with connected compact supp ort and a finite num ber of zeros in the supp ort. If the densit y is locally b ounded below b y some common pow er | x − x 0 | b − 1 around eac h of the zeros x 0 , its asymptotic error rate is at least that of the p o w er-la w in question. Conv ersely , if the densit y is lo cally b ounded ab o v e by some pow er-la w | x − x 0 | a − 1 in the neighborho o d of some arbitrary zero, the asymptotic order of conv ergence cannot exceed that of the corresp onding p o w er-la w. In fact, for the latter prop ert y , it suffices to consider later al neigh b orhoo ds, i.e. ( x 0 − ε, x 0 ] or [ x 0 , x 0 + ε ). 3.3. The partial transp ort case. In man y cases, one is in fact in terested in the riskiest p art of the riskiest dep endence structure, rather than in the whole dep endence structure. Mathematically , this translates into a sp ectral function with non-full supp ort (1 − m, 1), so that only the most dangerous fraction of prescrib ed mass m ∈ (0 , 1) is tak en into account in the sp ectral risk measure. This section is devoted to the sp ecific case where the sp ectral risk measure in question is the conditional v alue at risk CV aR m , corresp onding to α ∝ 1 (1 − m, 1) . Since α v anishes on the interv al (0 , 1 − m ) and is constant on (1 − m, 1), Problem ( P ) has the follo wing p artial m ultimarginal optimal transp ort form ulation max γ ∈ Γ m ( ρ 1 ,...,ρ D ) 1 m ˆ X c dγ , ( P m ) where Γ m ( ρ 1 , . . . , ρ D ) denotes the set of (p ositiv e) measures of mass m on X whose marginals are resp ectiv ely dominated by the ρ j . That is, a measure γ ∈ M + ( X ) is in Γ m ( ρ 1 , . . . , ρ D ) 14 ADRIEN CANCES, QUENTIN M ´ ERIGOT, AND LUCA NENNA if and only if we hav e γ ( X ) = m and for every j ∈ { 1 , . . . , D } , [ π j # γ ]( B ) ≤ ρ j ( B ) , ∀ Borel set B ⊆ X j . (3.20) W e refer to the work of Kitagaw a and Pass in [9] for more details on partial multimarginal transp ort. As a consequence, we lo ok for a discrete measure of the form δ m Y := mδ Y with Y ∈ X N , and the corresp onding discretized problem reads min Y ∈ X N − 1 N N X i =1 c ( y i ) + λ N D X j =1 W p, max ( ρ j , π j # δ Y ) p , ( P m N ) where W p, max denotes the p artial p -Wasserstein c ost . F or t w o (positive) measures ρ, ν ∈ M + p ( R n ) with finite p -moment, the latter is defined by W p, max ( ρ, µ ) p := min ζ ∈ Γ max ( ρ,ν ) ˆ R n × R n | x − y j | p dζ ( x, y ) , (3.21) where Γ max ( ρ, ν ) is the set of measures on R n × R n of mass min { ρ ( R n ) , ν ( R n ) } whose marginals are resp ectiv ely dominated b y ρ and ν . The subscript max refers to the fact that the maximum of quantit y is transp orted b et w een the tw o measures. Unlike the usual W asserstein distance, the partial cost W p, max is not a distance in the mathematical sense, as it lacks the separation prop ert y . Indeed, it is straightforw ard to show that W p, max ( ρ, ν ) is zero if and only if one of ρ or ν is dominated by the other one. Since the discrete measure in ( P m N ) is of mass m < 1, the p enalt y terms p enalize ho w far its marginals are from b eing resp ectiv ely dominated b y the ρ j , which is precisely what we desire to take in to account the inequalit y constraints (3.20) via the p enalt y metho d. F or the con v ergence result of this section, we will use the follo wing assumption. Assumption 2. We have p ∈ [1 , ∞ ) , and ther e exists β ∈ (0 , 1] such that c is β -H¨ older, with H¨ older c onstant C H . The follo wing lemma is a direct consequence of the ab o v e assumption. Of course, the estimate in question is deducible from Lemma 10 by noticing that α m ∝ 1 (1 − m, 1) satis- fies Assumption 1 with q = ∞ , but we prefer to pro vide a separate pro of for the sak e of completeness. Lemma 25 (H¨ older contin uit y of the CV aR ob jective function) . Under Assumption 2, for any p air of me asur es γ , ˜ γ on X of mass m , we have     1 m ˆ X c dγ − 1 m ˆ X c d ˜ γ     ≤ C H W p ( γ , ˜ γ ) β . (3.22) Pr o of. F or any coupling Π ∈ Γ( m − 1 γ , m − 1 ˜ γ ) ⊆ P ( X × X ), the left-hand side of (3.22) is b ounded ab ov e b y     ˆ X ( c ( x ) − c ( ˜ x )) d Π( x, ˜ x )     ≤ ˆ X × X | c ( x ) − c ( ˜ x ) | d Π( x, ˜ x ) ≤ C H ˆ X × X | x − ˜ x | β d Π( x, ˜ x ) , where we used β -H¨ older con tin uity of c . T aking the infimum ov er Π and using the standard inequalit y W β ≤ W p deriv ed from β ≤ p yields the desired estimate. □ Lemma 26. F or any two me asur es γ , ˜ γ ∈ M + p ( X ) with finite p -moment and of same finite mass, we have W p ( γ , ˜ γ ) p ≳ D X j =1 W p, max ( π j # γ , π j # ˜ γ ) p . (3.23) P AR TICLE METHOD FOR A NONLINEAR MUL TIMARGINAL OT PROBLEM 15 Pr o of. T ake an arbitrary joint measure Π ∈ Γ( γ , ˜ γ ), of mass m b y definition. Thanks to equiv alence of norms in R D , we find ˆ X × X ∥ x − ˜ x ∥ p d Π( x, ˜ x ) ≳ D X j =1 ˆ | x j − ˜ x j | p d [ π j,j + D # Π]( x j , ˜ x j ) . Since π j,j + D # Π is a joint measure of the respective j -th marginals of γ and of ˜ γ , the j -th term in the sum is b ounded b elo w by W p ( π j # γ , π j # ˜ γ ) p , and the desired inequality is obtained by taking the infimum in Π. □ As in the previous section, we introduce the functionals F m , F m N : M + p ( X ) → R ∪ { + ∞} that are implicitly minimized in Problems ( P m ) and ( P m N ), resp ectiv ely . These are defined b y ( F m ( γ ) = − 1 m ´ X c dγ + χ Γ m ( ρ 1 ,...,ρ D ) ( γ ) , F m N ( γ ) = − 1 m ´ X c dγ + λ N P D j =1 W p, max ( ρ j , π j # γ ) p + χ D m N ( γ ) , where D m N = { mδ Y : Y ∈ X N } is the set of uniform clouds of N p oints in X that are of mass m . By the same argumen ts as in the b eginning of Section 3.1, with the additional inequality W p, max ( ρ j , mδ Y j ) ≤ W p ( π j # γ ∗ , mδ Y j ), we find | min F m N − min F m | = O ( u β N ) , where u N is the optimal N -p oin t uniform quantization error of some solution γ ∗ . The con- struction of γ N is detailed in the next lemma, and is essentially the same as in Section 3.1, except that we consider the p artial optimal transp orts from the ρ j to the resp ectiv e pro jec- tions of the p oin t cloud. Lemma 27 (Blo c k appro ximation) . L et mδ Y ∈ D N b e a uniform cloud of mass m c onsisting of N p oints in X , and denote mδ Y j its pr oje ction on X j . Ther e exists a me asur e γ ∈ Γ m ( ρ 1 , . . . , ρ D ) such that W p ( γ , mδ Y ) ≲   D X j =1 W p, max ( ρ j , mδ Y j ) p   1 p . (3.24) Pr o of. Consider the activ e parts ρ act j of the probability measures ρ j in their resp ectiv e W p - optimal transp ort to the pro jections mδ Y j . That is, the partial optimal transp ort plan for W p, max ( ρ j , mδ Y j ) has first marginal ρ act j . Applying Lemma 12 to δ Y and the normalized measures m − 1 ρ act j , we obtain some probabilit y measure γ prob ∈ Γ( m − 1 ρ act 1 , . . . , m − 1 ρ act D ) satisfying W p ( γ prob , δ Y ) ≲   D X j =1 W p ( m − 1 ρ act j , δ Y j ) p   1 p . Letting γ := mγ prob ∈ Γ m ( ρ act 1 , . . . , ρ act D ) ⊆ Γ m ( ρ 1 , . . . , ρ D ) then yields the desired estimate, thanks to the scaling W p ( mµ, mν ) = m 1 p W p ( µ, ν ) and to the equality W p ( ρ act j , mδ Y j ) = W p, max ( ρ j , mδ Y j ), b y optimality of the ρ j . □ Lemma 12 leads to the following global con vergence results for the partial version of the problem. Theorem 28. Supp ose that Assumption 2 holds, and let u N b e some upp er b ound on the W p -optimal uniform quantization err or for some fixe d minimizer γ ∗ of F . Then, letting λ N = u − ( p − β ) N , we have | min F m N − min F m | = O ( u β N ) . (3.25) 16 ADRIEN CANCES, QUENTIN M ´ ERIGOT, AND LUCA NENNA Mor e over, for any se quenc e of minimizers δ Y ∗ N ∈ argmin F m N , N ∈ N , every we ak limit p oint is a minimizer of F m , and for every j ∈ { 1 , . . . , D } we have W p, max ( ρ j , π j # δ Y ∗ N ) = O ( u N ) . (3.26) Once again thanks to the direct generalization of M ´ erigot and Mireb eau’s result stated in Prop osition 16, we deduce the following immediate corollary . Corollary 29. Supp ose that Assumption 2 holds and that the ρ j ar e al l c omp actly supp orte d. L et d ∈ [1 , D ] b e the b ox dimension of the supp ort of some minimizer of F . Then, letting λ N = τ p,d ( N ) − ( p − β ) wher e τ p,d is define d in (3.12) , we have | min F m N − min F m | = O        N − β d if d > p, (log N ) β p N − β p if d = p, N − β p if d < p, (3.27) wher e O hides a c onstant which dep ends on the supp ort of the minimizer in question. Mor e- over, for any se quenc e of minimizers mδ Y ∗ N ∈ argmin F m N , N ∈ N , every we ak limit p oint is a minimizer of F m . F or a result analogous to Theorem 24, one needs to control the uniform quan tization error of submeasures of the form ( F − 1 ρ ) # L (1 − m, 1) in terms of that of ρ , for sufficien tly w ell-b eha v ed univ ariate probability measures ρ . Lemma 30 (Quantization of the restriction to an in terv al) . L et ρ ∈ P p ( R ) b e absolute c ontinuous with c onne cte d supp ort. Supp ose that, as N → ∞ , e p,N ( ρ ) ≲ (1 + log N ) ζ N η for some exp onents η > 0 and ζ ∈ R , wher e ≲ hides a c onstant which may dep end on ρ and on the exp onents. Then, for any m ∈ (0 , 1) , the me asur e ρ m := ( F − 1 ρ ) # L (0 ,m ) of mass m satisfies e p,N ( ρ m ) ≲ (1 + log N ) ζ N η (3.28) as N → ∞ , and the c onstant b ehind ≲ is indep endent of m . The same estimate holds for the me asur e ˜ ρ m := ( F − 1 ρ ) # L (1 − m, 1) . Remark 31. The assumption that m is r ational makes the pr o of r ather str aightforwar d, but is likely unne c essary, as suggeste d by the multiplic ative c onstant b ehind ≲ in (3.28) b eing indep endent of m . However, one c annot dir e ctly extend our r esult to any arbitr ary m . Inde e d, and as highlighte d in the pr o of, for a r ational m = a b with a, b c oprime p ositive inte gers, the estimate (3.28) only holds for N ≥ a . Pr o of. Denote u ( N ) = (1+log N ) ζ N η , and write m = a b with a < b coprime p ositive in tegers. Let ( bK ) − 1 P bK i =1 δ x i b e an optimal uniform quan tizer of ρ for W p , where w e suppose without loss of generality that x 1 < x 2 < · · · < x bK . Since m ( bK ) = aK is an integer, the W p -optimal transp ort from ρ to its uniform quantizer exactly send the measure ρ m to the leftmost aK Dirac masses. As a result, e p,aK ( ρ m ) ≤ W p ( ρ m , ( bK ) − 1 P aK i =1 δ x i ) ≤ W p ( ρ, ( bK ) − 1 P bK i =1 δ x i ) = e p,bK ( ρ ) ≤ e p,aK ( ρ ) , where the last inequality comes from the straigh tforw ard observ ation that for an absolutely con tin uous measure on R with c onne cte d supp ort, the uniform quantization error is decreas- ing in the num b er of Dirac masses. W e thus hav e e p,N ( ρ m ) ≤ e p,N ( ρ ) for any N that is a P AR TICLE METHOD FOR A NONLINEAR MUL TIMARGINAL OT PROBLEM 17 m ultiple of a . No w consider an arbitrary N , and denote N = aK + r its Euclidean division b y a . Thanks to ρ m ha ving connected supp ort as well, e p,N ( ρ m ) ≤ e p,aK ( ρ ) = u ( aK ) = u ( aK ) u ( aK + r ) u ( aK + r | {z } N ) . Since u ( n ) = (1+log n ) ζ n η and r ∈ { 0 , 1 , . . . , a − 1 } , elemen tary considerations sho w that the ratio u ( aK ) u ( aK + r ) is b ounded b y 2 η (1 + log 2) max {− ζ , 0 } as so on as K ≥ 1, so for any N ≥ a , whic h concludes the pro of. □ Theorem 32. Supp ose that Assumption 2 holds, that m ∈ (0 , 1) is a r ational, and that the c ost function c is sup ermo dular, and monotone incr e asing in e ach variable. Supp ose mor e over that the mar ginal pr ob ability me asur es ρ j ar e absolutely c ontinuous with c onne cte d supp orts, and that ther e exists exp onents η > 0 , ζ ∈ R such that for every j ∈ { 1 , . . . , D } , e p,N ( ρ j ) ≲ h N := (1 + log N ) ζ N η , wher e ≲ hides a c onstant which may dep end on ρ j and on the exp onents η , ζ . Then, letting λ N = h − ( p − β ) N , we have | min F m N − min F m | = O ( h β N ) . (3.29) Mor e over, for any se quenc e of minimizers δ Y ∗ N ∈ argmin F m N , N ∈ N , every we ak limit p oint is a minimizer of F m . 4. Numerics W e now illustrate our discretization metho d with v arious numerical simulations using corresp onding to the differen t cases dealt with in the previous section. Eac h simulation is made with some fixed n umber of p oin ts N , and we compute a numer- ical solution of the corresp onding Problem ( P N ) or ( P m N ) using the Limited-memory BF GS algorithm pro vided by SciPy . The latter, a quasi-Newton metho d, only requires the v alue of the optimized functional and that of its gradien t at the curren t p oin t. W e ha ve already seen that computing the v alue and gradien t of the sp ectral risk measure R α ( c # δ Y ) is immediate, since this quan tit y is a linear com bination of the c -v alues at the p oin ts of the cloud, and since the cost function c is explicit. The computation of the p enalt y terms and of their gradien ts is a bit more inv olv ed, as it inv olv es solving a semi-discrete optimal transp ort problem, partial or not dep ending on the c hosen framew ork, for eac h marginal. In this pap er, w e fix p = 2 and restrict ourselv es to unidimensional marginals for the simulations, which considerably eases the computation related to the p enalt y terms. The initial p oint cloud Y init is constructed b y drawing N independent p oin ts from the uniform measure on the pro duct of the supp orts spt µ j . Remark 33 (General rule of thum b for the p enalt y co efficien t) . We observe that when the p enalty c o efficient is qualitatively lar ge, the pr oje ctions of the numeric al solution on e ach of the axes match wel l with the r esp e ctive pr escrib e d mar ginals ρ j , but the p oint cloud in question lacks structur e, in the sense that the input variables ar e p o orly c orr elate d. On the other hand, a p enalty c o efficient that is qualitatively to o smal l le ads to a much cle ar er structur e, but at the c ost of a p o or appr oximation of the mar ginals. T o addr ess this issue, we start by solving the discr etize d pr oblem with a r e asonably smal l p enalty c o efficient, in or der to get the r elevant structur e for the solution, and gr adual ly incr e ase the c o efficient until the pr oje ctions on the differ ent axes ar e satisfyingly close to the pr escrib e d mar ginals. A lthough deriving a universal rule for this se quenc e of p enalty c o efficients would likely b e quite delic ate, this br o ad str ate gy c an serve as a gener al rule of thumb to c ompute satisfying numeric al solutions. 18 ADRIEN CANCES, QUENTIN M ´ ERIGOT, AND LUCA NENNA 4.1. Computing the p enalt y terms. Let us first giv e the expression of the resp ectiv e gradien ts of the unidimensional semi-discrete optimal transp ort functional G : Z ∈ R N 7→ W 2 ( ρ, δ Z ) 2 and of its partial version G m : Z ∈ R N 7→ W 2 , max ( ρ, mδ Z ) 2 , where ρ ∈ P 2 , ac ( R ) is some fixed univ ariate probability density with finite v ariance and where m ∈ (0 , 1). Once again, we write Z = ( z 1 , . . . , z N ). Since G and G m are only differen tiable outside the general- ized diagonal, we assume that the z i are pairwise distinct, so that without loss of generalit y z 1 < · · · < z N . F or the sak e of conciseness, w e only deriv e the gradien t of G m , as the gradien t of G can b e derived similarly . Let us consider the dual formulation G m ( Z ) = max ψ ∈ (0 , + ∞ ) N ( ˆ R min { 0 , min i [ | x − z i | 2 − ψ i ] } dρ ( x ) + m N N X i =1 ψ i ) , (4.1) whic h stems from the Kan toro vic h duality for partial optimal transport. W e refer to [15, Theorem 1.42] for the duality theory of standard optimal transp ort, and to [2] for the case of partial transp ort. Note that b y dominated con v ergence, the optimalit y condition on ψ is that ρ (RLag i ( Z ; ψ )) = m N for all i , where RLag i ( Z ; ψ ) = Lag i ( Z ; ψ ) ∩ B ( x i , p ψ ∗ i ) is the r estricte d L aguerr e c el l corresp onding to ψ . Here, Lag i ( Z ; ψ ) , i ∈ { 1 , . . . , N } denote the cells of the (standard) L aguerr e tessel lation induced by ψ : Lag i ( Z ; ψ ) = { x ∈ R : | x − y i | 2 − ψ i < | x − y k | 2 − ψ k , ∀ k  = i } . (4.2) Let us denote RLag m i ( ρ, Z ) the restricted Laguerre cells induced by the (unique) optimal ψ for the partial optimal transp ort from ρ to mδ Z . The env elope theorem then yields ∂ G m ∂ z i ( Z ) = 2 ˆ R Lag m i ( ρ,Z ) ( z i − x ) dρ ( x ) = 2 m N [ z i − b m i ( ρ, Z )] , (4.3) where b m i ( ρ, Z ) = ffl RLag m i ( ρ,Z ) x dρ ( x ) is the ρ -barycenter of the optimal restricted cell cor- resp onding to z i . The gradient of G is derived in the exact same wa y but via the standard Kan toro vich duality , and we hav e ∂ G ∂ z i ( Z ) = 2 N [ z i − b i ( ρ, Z )] , (4.4) with b i ( ρ, Z ) = ffl Lag i ( ρ,Z ) x dρ ( x ) the ρ -barycenter of the optimal (unrestricted) Laguerre cell corresp onding to z i . In the balanced case, the optimal Laguerre tessellation is simply given by the uniform decomp osition of ρ into N equal-mass parts with resp ectiv e supp orts m utually ordered, namely Lag i ( ρ, Z ) = ( F − 1 µ ( i − 1 N ) , F − 1 µ ( i N )). Note that the latter only dep ends on the or der of the Dirac p ositions. In the partial transp ort case, this is usually not the case anymore, since, broadly sp eaking, the “gaps” corresp onding to the mass that is not transp orted will roughly b e lo cated b et ween the pairs of consecutive cells whose asso ciated Dirac masses are far enough from each other, see Figure 2. T o solve (4.1), we use a co de dev elop ed by Hugo Leclerc, in which the restricted Laguerre cells (whic h are interv als) are directly parameterizing b y their resp ective endp oin ts. The co de relies on a rather efficien t heuristic that successively merges well-c hosen adjacent cells in order to reduce the problem’s complexit y and num b er of v ariables. 4.2. Squared sum of co ordinates. In [13, Remark 2.12], P ass men tions a no w w ell-known example of a cost function for which solutions of the standard m ultimarginal optimal trans- p ort problem with D marginals may ha ve supp ort of dimension D − 1. The surplus cost function in question is c ( x 1 , . . . , x D ) = −| x 1 + · · · + x D | 2 , and indeed it is immediate to chec k that any probability measure concen trated on the plane x 1 + · · · + x D = 0 is optimal for P AR TICLE METHOD FOR A NONLINEAR MUL TIMARGINAL OT PROBLEM 19 (a) Balanced transp ort (b) Partial transp ort, with m = 1 2 Figure 2. W e represen t the decomposition of the unidimensional probability densit y induced by its optimal transp ort to the sum of Dirac masses, whose resp ectiv e positions are indicated b y the v ertical lines. The left-hand side corresp onds to s tandard optimal transp ort, with the Dirac weigh ts summing to one, while the right-hand corresp onds to partial optimal transp ort, with the Dirac weigh ts summing to m = 1 2 . its marginals. In fact, fixing univ ariate probabilit y measures µ 1 , . . . , µ D ∈ P ( R ), Jensen’s inequalit y yields that for any γ ∈ Γ( µ 1 , . . . , µ D ) we hav e ˆ R D | x 1 + · · · + x D | 2 dγ ≥  ˆ R D ( x 1 + · · · + x D ) dγ  2 =   D X j =1 E µ j   2 . Hence any coupling of the µ j that is concentrated on the hyperplane x 1 + · · · + x D = P j E µ j m ust b e optimal. Our numerical sim ulations in Figure 3 sho w suc h an optimal coupling with tw o-dimensional supp ort for µ 1 the uniform measure on [0 , 2], µ 2 the triangle density T ri(0 , 1 , 2) and µ 3 the Wigner semicircle densit y with support [ − 1 , 1]. The blac k v ector vector in the 3D graphs has base-p oin t ( E µ 1 , E µ 2 , E µ 3 ) = (1 , 1 , 0) and is collinear with (1 , 1 , 1). Figure 3. Numerical solution for the standard multimarginal problem with three marginals and surplus cost c ( x 1 , x 2 , x 3 ) = −| x 1 + x 2 + x 3 | 2 . The three marginal densities are shown in the first ro w, o verlapped with the resp ectiv e histograms of the corresp onding pro jections of our numerical solution. The differen t 3D views in the second row sho w that, as expected, the solution concen trates on the plane of equation x 1 + x 2 + x 3 = 0. W e used N = 3 000 p oin ts and the sequence of p enalt y coefficients λ ∈ 10 k : k = − 2 − , − 2 , . . . , 4 } . 20 ADRIEN CANCES, QUENTIN M ´ ERIGOT, AND LUCA NENNA (a) Partial transp ort plan, m = 0 . 5 (b) Partial barycenter, m = 0 . 5 (c) Partial transp ort plan, m = 0 . 75 (d) Partial barycenter, m = 0 . 75 Figure 4. Numerical simulation for Kitagaw a and P ass’ example in the pro of of [9, Prop osition 4.1], which sho ws non-monotonicit y of the partial barycen ter (in green) in its mass m . The blue and orange densities corre- sp ond to the t w o probability marginals, with their activ e parts dark ened. Both for these active parts and for the partial barycenter, w e use a simple k ernel densit y estimation to conv ert the corresp onding p oin t clouds obtained n umerically into densities. The v alue ε inv olv ed in the marginal densities defined b y the aforemen tioned authors w as set to 1 3 . W e used N = 1 000 p oin ts, equal weigh ts λ 1 = λ 2 = 1 2 , and the sequence of p enalt y co efficients λ ∈ { 10 k : k = − 1 − , 0 , . . . , 4 } . 4.3. W asserstein barycen ter. The multimarginal form ulation of the partial W asserstein barycen ter problem reads min γ ∈ Γ m ( ρ 1 ,...ρ D ) ˆ R D D X j,k =1 λ j λ j | x j − x k | 2 dγ ( x ) . (4.5) In Figure 4, w e repro duce numerically the example given by Kitagaw a and Pass to prov e [9, Prop osition 4.1]. Their example illustrates non-monotonicity of the partial barycenter in its mass m , even for D = 2 measures. It turns out that our discretization do es not p erform w ell on the partial barycenter prob- lem, as it tends to fall into nonlo cal minima. This issue is illustrated in Figures 5 and 6, whic h b oth corresp ond to the partial barycenter computation of D = 2 measures, for the sak e of simplicit y . The n umerical solution in Figure 5 is fine, since it selects the common part of b oth marginals, which has exactly the prescrib ed mass. Also, note that the underlying partial transport plan is indeed concen trated on the line x 1 = x 2 . Figure 6 sho ws another ex- ample, in whic h w e once again prescribe the mass of the barycen ter to b e that of the common part of the t wo densities. Ev en though the n umerical solution still has has one-dimensional supp ort, it misses some of the common mass and selects it elsewhere, yielding a strictly p ositiv e total cost. This raises the question of the existence of a critical p oin t as N → ∞ . As men tioned ab o v e, we initialize the Limited-memory BF GS algorithm by taking a cloud of p oin ts indep enden tly drawn from the uniform measure on the pro duct of the supp orts of the marginals. P erhaps a more astute initialization would allow for b etter n umerical solutions of our discretized partial barycenter problem, but it remains to b e found. P AR TICLE METHOD FOR A NONLINEAR MUL TIMARGINAL OT PROBLEM 21 (a) Partial barycenter (b) Active submeasures (c) Partial transp ort plan Figure 5. Numerical solution for the multimarginal formulation of the par- tial barycen ter problem b et w een the triangle densit y T ri(0 , 1 , 2) and its trans- lation by 0 . 7. W e used N = 1 000 p oin ts, mass m = 0 . 4225, equal weigh ts λ 1 = λ 2 = 1 2 , and the sequence of p enalt y co efficien ts λ ∈ { 10 k : k = − 3 − , − 2 , . . . , 2 } . The chosen mass is exactly equal to the amount of common mass of the tw o marginal probability measures. The orange point cloud in 5 c represen ts the partial transp ort plan w e find whereas the green line is the standard optimal transp ort plan b et w een the tw o marginals. (a) Partial barycenter (b) Active submeasures (c) Partial transp ort plan Figure 6. Numerical solution for the multimarginal formulation of the par- tial barycen ter problem b et w een the triangle densit y T ri(0 , 1 , 2) and the densit y 2 3 T ri( − 1 , 0 , 1) + 1 3 T ri(1 , 2 , 3). W e used N = 1 000 p oin ts, mass m = 0 . 4225, equal w eigh ts λ 1 = λ 2 = 1 2 , and the sequence of p enalt y co- efficien ts λ ∈ 10 k : k = − 3 − , − 2 , . . . , 6 } . The chosen mass is exactly equal to the amoun t of common mass of the tw o marginal probability measures, and in particular, we see that the n umerical solution is far from optimal. 4.4. Repulsiv e cost. The standard m ultimarginal optimal transp ort problem with Coulom b cost c and univ ariate marginal ρ ∈ P ( R ) reads min γ ∈ Γ D ( ρ ) ˆ R D c ( x 1 , . . . , x D ) dγ ( x 1 , . . . , x D ) , (4.6) where c ( x 1 , . . . , x D ) = X 1 ≤ j

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