The Unbalanced Gromov Wasserstein Distance: Conic Formulation and Relaxation
Authors: Thibault Sejourne
François-Xavier Vialard
Gabriel Peyré
Venue:  (2021)
Keywords: Gromov-Wasserstein
Non-convex optimization
Optimal Transport
Partial matching

Abstract: Comparing metric measure spaces (i.e. a metric space endowed with a probability distribution) is at the heart of many machine learning problems. This includes for instance predicting properties of molecules in quantum chemistry or generating graphs with varying connectivity. The most popular distance between such metric measure spaces is the Gromov-Wasserstein (GW) distance, which is the solution of a quadratic assignment problem. This distance has been successfully applied to supervised learning and generative modeling, for applications as diverse as quantum chemistry or natural language processing. The GW distance is however limited to the comparison of metric measure spaces endowed with a probability distribution. This strong limitation is problematic for many applications in ML where there is no a priori natural normalization on the total mass of the data. Furthermore, imposing an exact conservation of mass across spaces is not robust to outliers and often leads to irregular matching. To alleviate these issues, we introduce two Unbalanced Gromov-Wasserstein formulations: a distance and a more tractable upper-bounding relaxation. They both allow the comparison of metric spaces equipped with arbitrary positive measures up to isometries. The first formulation is a positive and definite divergence based on a relaxation of the mass conservation constraint using a novel type of quadratically-homogeneous divergence.This divergence works hand in hand with the entropic regularization approach which is popular to solve large scale optimal transport problems. We show that the underlying non-convex optimization problem can be efficiently tackled using a highly parallelizable and GPU-friendly iterative scheme. The second formulation is a distance between mm-spaces up to isometries based on a conic lifting. Lastly, we provide numerical simulations to highlight the salient features of the unbalanced divergence and its potential applications in ML.

Under review as a conference paper at ICLR 2021
THE UNBALANCED GROMOV WASSERSTEIN DIS- TANCE: CONIC FORMULATION AND RELAXATION
Anonymous authors
Paper under double-blind review
ABSTRACT
Comparing metric measure spaces (i.e. metric space endowed with probability distribution) is at the heart of many machine learning problems. This includes for instance predicting properties of molecules in quantum chemistry or generat- ing graphs with varying connectivity. The most popular distance between such metric measure spaces is the Sainoo--Wsssesteinnn (GW) distance, which is the solution of a quadratic assignment problem. This distance has been successfully applied to supervised learning and generative modeling, for applications di- verse as quantum chemistry or natural language processing. The GW distance is however limited the comparison of metric measure spaces endowed with a probability distribution. This strong limitation is problematic for many applica- tions MI. where there is no priori natural normalization on the total mass of the data. Furthermore, imposing exact conservation of mass across spaces is not robust to outliers and often leads to irregular matching. To alleviate these is- sues, we introduce two Unbalanced Gromov- Wasserstein formulations: distance and a more tractable upper- bounding relaxation. They both allow the comparison of metric spaces equipped with arbitrary positive measures up isometries. The first formulation is a positive and definite divergence based on a relaxation of the mass conservation constraint using a novel type of quadratically- -omooggeooss divergence. This divergence works hand in hand with the entropic regularization approach which is popular solve large scale optimal transport problems. We show that the underlying non-convex optimization problem can be efficiently tack- led using highly parallelizable and GPU-friendly iterative scheme. The second formulation is a distance between mm-spaces up to isometries based on a conic lifting. Lastly, we provide numerical simulations to highlight the salient features of the unbalanced divergence and its potential applications in ML.
INTRODUCTION
Comparing data distributions on different metric spaces a basic problem in machine learning- This class of problems is for instance the heart of surfaces (Bronstein al.) (2066) graph matching (Xu et al. 2019) (equipping the surface or graph with its associated geodesic distance), regression problems in quantum chemistry (Gilmer et al. 2017) (viewing the molecules as dis- tributions of points in R²') and natural language processing (Grave et al. 2019 Alvarez-Melis & Jaakkola 2018) (where texts in different languages are embedded as points distributions different vector spaces),
Metric measure spaces. The mathematical way to formalize these problems is to model the data as metric measure spaces (mm-spaces). mm-space is denoted as X' (X, d. p)) where X is a complete separable set endowed with distance d and positive Borel measure μ (X). For instance, if X = (æ+); is finite set of points, then μ = E; m&dæ, (here òg, is the Dirac mass at x;) simply set of positive weights m, (({x))) }) 2 0 associated to each point x;, which accounts for its mass or importance, For instance, setting some m; to 0 is equivalent to removing the point xy- We refer to Sturm (2012) for mathematical account on the theory of mm-spaces.
In all the applications highlighted above, makes sense to perform the comparisons up to iso- metric transformations of the data. Two mm-spaces X' (X,dx, p) and y (Y,yy

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considered to be equal (denoted X' 3) if they are isometric, meaning that there is bijection & spt(H) + spt(y) (where spt(μ) is the support of (4) such that dx (z. y) = dy (v(z), (t)) is equivalent to imposing v(A) (A) for any set A C Y. For discrete spaces where μ E; midu, then one should have v = Vi = E; m.do(z,). As highlighted by Mémoli (2011). considering mm-spaces up to isometry is powerful way to formalize and analyze wide vaniey of problems such matching, regression and classification of distributions of points belonging to different spaces, The key to unlock all these problems is the computation of a distance between mm-spaces up isometry. So far, existing distances (reviewed below) assume that /s is probability distribution, i.e. ju() 1. This constraint is not natural and sometimes problematic for most of the practical applications to machine learning. The goal of this paper is to alleviate this restriction. We define for the first time a class of distances between unbalanced metric measure spaces, these distances being upper-bounded by divergences which can be approximated by an efficient numerical scheme.
and dh = v. Here by is the push-forward operator,
that 2/
Csiszár divergences The simplest case is when X Y and one simply ignores the underly- ing metric. One can then use Csiszár divergences (or g-divergences), which perform a point- wise comparison (in contrast with optimal transport distances, which perform a displacement comparison). It is defined using an entropy function yp R+ + [0, +oo], which is con- vex, lower semi-continuous, positive function with (o(1) = O. The Csiszár p-divergence reads (ulu) def. I (((t) (=))d=  + +y'o fx dut,, where μ du
de + -- is the Lebesgue decomposition
of with respect to v and &oc lim,- ((r)/r E IR U [+00] is called the recession con- stant. This divergence D, is convex, positive, 1-homogeneous and weak* lower-semicontinuous see Liero et al. (2015) for details. Particular instances of y-divergences are Kullback-Leibler (KL)
for (p() = r log(r) +1 I (note that 4 = oo) and Total Variation (TV) for ((r)= = |r - 1/.
Balanced and unbalanced optimal transport. If the common embedding space X is equipped with a distance d(x y), one can use more elaborated methods such as optimal transport (OT) dis- tances, which are computed by solving convex optimization problems. This type of methods has proven useful for ML problems as diverse as domain adaptation (Courty et al. 2014), supervised learning over histograms (Frogner et al. 2015) and unsupervised learning of generative models (Ar- jovsky et al.] 2017). In this case, the extension from probability distributions to arbitrary positive measures (p,v) (X) is now well understood and corresponds to the theory of unbalanced OT. Following Liero et al. (2015): Chizat et al. (2018c), a family of unbalanced Wasserstein dis-
tances is defined by solving
(1) Here (TT, are the two marginals of the joint distribution π, defined by тI (A) = T(A x Y) for R and exponent g 2 1 should be chosen wisely to ensure for instance that UW defines a distance (see Section 2.2.1 lp. It is frequent to take pD. instead of D (i.c. take v0 = ppp) to adjust the strength of the marginals' penalization. Balanced OT is retrieved with the convex indicator & 441 or by taking the limit p - +00, which enforces म1 = /u and π. v. When 0- < p < +00, unbalanced OT operates a trade-off between transportation and creation of mass, which is crucial to be robust to outliers in the data and cope with mass variations in the modes of the distributions. For supervised tasks, the value of should be cross-validated to obtain the best performances.
UW( del
inf nEM(xxx)
f A(d(z, v)) v) + (T(()) + (T2ll))))
A C X. The mapping À= R+ -
Its use is gaining popularity in applications, such as medical imaging registration (Feydy et al. 2019ap, videos (Lee al., 2019). generative learning (Balaji et al.) 2020) and gradient flow to train neural networks (Chizat & Bach, 2018. Rotskoff et al. 2019). Furthermore, existing efficient algorithms for balanced OT extend to this unbalanced problem. In particular Sinkhorn's 's iterations, introduced in ML for balanced OT by Cuturi (2013), extend to unbalanced OT (Chizat et al., 201 8a Séjourné et al. 2019), as detailed in Section
The Gromov- Wastov--esserttein distance and its applications. The Gromov- Wasseerttenn (GW) dis- tance (Mémoli 2011 Sturm 2012) generalizes the notion of OT the setting of mm-spaces up to isometries. It corresponds to replacing the linear cost f X(d)dx of OT by a quadratic function
(XX)) { A(lxx(x; (x, - dy (v. o/)dd (dd .y')
GW(X)))) dt min
m2 v/ (2)
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It is proved in Mémoli 2011
Sturm (2012) that GW defines with X(t) t distance up to isometries on balanced mm sam-spaces (i.e. the measures are probability distributions). In this paper, we extend this construction to arbitrary positive measures, and provide explicit settings in Section 2.2.1
This distance is applied successfully in natural language processing for unsupervised translation learning (Grave et al., 2019 Alvarez-Melis & Jaakkola 2018), in generative learning for objects lying in spaces of different dimensions (Bunne et al. 2019 and to build VAE for graphs (Xu et al.) 2020). It has been adapted for domain adaptation over different spaces (Redko et al. (0200) It is also a relevant distance to compute barycenters between graphs or shapes (Vayer et al. 2018 @018)Chowdhur
& Needham 2020). When (X, 3) are Euclidean spaces, this distance compares distributions up to
rigid Isometry, and is closely related (but not equal) to metrics defined by procrustes analysis (Grave
ct al. 2019 Alvarez-Melis et al. 2019).
The problem is non convex because the quadratic form I A(ldx - ldw @ is not positive in general. It is in fact closely related to quadratic assignment problems (Burkard al. 1998), which are used for graph matching problems, and are known to be NP-hard in general. Nevertheless, non- convex optimization methods have been shown to be successful in practice to use GW distances for ML problems. This includes for instance alternating minimization (Mémoli 2011 Redko et al., 2020) and entropic regularization (Peyré et al. 2016 Gold & Rangarajan 1996).
The concomitant work Df Ponti & Mondino 2020) extends the L" transportation distance defined in Sturm et al. (2006) to unbalanced mm-spaces and studies its geometrie properties. This distortion distance is not equivalent to the GW distance, and is more difficult to estimate numerically because it explicitly imposes triangle inequality constraint in the optimization problem. The work of Chappl et al. (2020) relaxes the GW distance to the unbalanced setting by hybridizing GW with partial OT (Figalli) 2010)) for unsupervised labeling. It ressembles one particular setting of our formulation, but with some important differences, detailed Section Our construction is also connected to partial matching methods, which find numerous applications in graphics and vision (Cosmo et al., 2016). In particular, Rodola et al. 2012) introduces a mass conservation relaxation of the GW problem.
Related works and contributions.
The two main contributions of this paper are the definition of two formulations relaxing the GW distance. The first one is called the Unbalanced Gromov-Wasserstee Wasserstein (UGW) divergence and can be computed efficiently on GPUs. The second one is called the Conic Kromov-Wasserstee distance be a distance between mm-spaces endowed with positive measures up to
(CGW). It is proved isometries, as stated in Theorem | which is the main theoretical result of this paper. We also prove in Theorem[ TireemTihi thht UGW can used a surrogate upper- upper-bounding bounding CGW. We present those concepts and their properties in Section 2 a particular setting of UGW. This method computes an approximate stationary point of the non- convex energy. It leverages the strength of entropic regularization and the Sinkhorn algorithm, namely that it is GPU-friendly and defines smooth loss functions amenable to back- propagationn for MI applications. Section 4 provides some numerical experiments to highlight the qualitative behavior of this algorithm, which shed some lights on the favorable properties of UGW to cope with outliers and mass variations in the modes of the distributions.
We also detail in Section 3 an efficient computational scheme for
UNBALANCED GROMOV- WANIOV--WSSSSSTENN FORMUL
We present this section our two new formulations and their properties. The first one, called UGW, is exploited in Sections 3 and to derive an efficient algorithm used in numerical experiments. The second one, called CGW, defines a distance between mm-spaces up to isometries. Those results build upon the results of Liero et (2015), and summary of the construction of UOT is detailed in Appendix A In all what follows, we consider complete separable mm-spaces endowed with a metric and positive measure.
2.1 THE UNBALANCED GROMOV- WASSERSTEIN DIVERGENCE
This new formulation makes of quadratic ---divergences, defined D (plp) di. (pppll)) where p & p E Mx33 is the tensor product measure defined by d(p @ p) y) = dp(z)dp(y). Note that D is not convex function in general.
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Definition 1 (Unbalanced GW). The Unbalanced Cromov-Wassersten Wasserstein divergence is defined as UGW(X, = inf (XXY ((T) where
((T)
def
MIdx(2,2) dy (v. ('))d(( y)dm(rl. D + (TTlp) + (Tell))) (3)
X2XY2
This definition can be understood as an hybridation between and (2) but with a twist: one needs to use the quadratic divergence De in place of D... To the best of our knowledge, it is the first time such quadratic divergences are being used and studied. In the TV case, this is the most important distinction between UGW and partial GW (Chapel et al.) 2020). Note also that the balanced GW distance
(2) is recovered as particular case when using entropy p.p.
t= or by letting p + +o0 for an
Using quadratic divergences results in UGW being 2- 2-homogenouus: for 0 2 O, writing (Na, Vo) equiped with (0μ, 0v),, one has 0 UGW )1o) UGW(X',)) When using tensorized p-divergences, the resulting unbalanced Gromov- Wassertein functional between 2g and Vo have very different and inconsistent behaviors when 0 0 and I +00. Indeed, once normalized by 0-2 and 0-1, one obtains respectively balanced GW and a Hellinger-type distance. Using tensorized divergences ensure that the behavior does not depends 0.
We first prove the existence of optimal plans т solution to (3). which holds for the three key settings of Section 2.2.T namely for KT, TV, and for compact metric spaces (such finite pointclouds and graphs) All proofs are deferred in Appendix B
Proposition 1 (Existence of minimizers)) We assume that (X, Y) are compact and that either (i) superlinear, ie &l 20, or (ii) X has compact sublevel sets in R.+, and 2. + inf X 0. Then there exists E (X x Y) such that UGW(X, 3) L(x).
The following proposition ensures that the functional UGW can be used to compare mm-spaces.
Proposition 2 (Definiteness of UGW). Assume that ({0)) = {1} and X ({0}) {0]. Then UGW(XX.)) 2 and if and only if X ~ V.
We end this section with reformulation of UGW (3) which is important to make the connection with the second formulation of the following section. Its proof is deferred to Appendix B It splits UGW into two parts: the term (0)) (I(1 + (v @ v)- accounts for the pure creation/destruution of mass and a new transport cost Th, accounts for the remaining part (partial/pure transport and partial creation/destruutiin of mass)
Lemma 1. Defining Le(a, b) del c ++ ap(1/a) + bp(1/b). and writing (f del. du del de the
dry 9 dwg
Lebesgue densities of (H, w.r. (m), such that μ fmi + +++ and v = g + vt,, one has
((T) = - Ex @ f.9 @ g) + +)
(4)
2.2
THE CONIC GROMOV- WASSERSTEIN DISTANCE
We introduce second "conic" formulation of unbalanced GW, which is connected to UGW, and whose construction is inspired by the conic formulation of UOT (see Appendix A for an overview).
2.2.1 1
BACKGROUND ON CONE SETS AND DISTANCES
The conic formulation lifts point x X to a couple (x,r) X x R+ where r encodes some (power of a) mass. Then seek optimal transport plans defined over C|X] X x R {0)), where coordinates (z, r = 0) with no mass are merged into single point ox called the apex of the cone. In the sequel, points of X R4 noted (xx, r). while [z.
quotiented points of c[X].
s']) While transport plans depend on variables ([æ, r]. [y, s)) and ([x',r"). [ in C[X] x C[Y], the transportation cost involved in conic formulation only makes of the 2-D cone C[R over R. endowed with the distance lu - o| (note that any other distance on R could be used as well). More specifically, we consider coordinates of the form ([u, (u, b))
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(1dx(x, ....a'))rr) (d. (v.3"), ss'I) E C[R+] x c[R+]. Thus now describe conic discrepancies
D on C[R,], which are defined for (p.9) 2 0 as
D[uu, a), (c, b))* det Hx(u-vl)( (a?, IP) where Ha(aP, (P)
def. inf ( 620
a 0
h"
is the perspective transform of Le introduced in Lemma The intuition underpinning the definition of this cost is that the perspective transform accounts for the possibility to rescale a transport plan by scalar but the scaling is performed pointwise instead of globally. In general 'D is not distance, but is always definite as stated by this result proved in Appendix A
Proposition 3. Assume A > ({0}) = (O), - ({0)) = {1} and yp is coercive. Then D is definite on C[R+], ie. D[[u, a), [t. b]) 0 and only if (a = 0) or (a = b and u v).
Of particular interest are those & where D is a distance, which necessitates a careful choice of X, and q. We now detail three examples where this is the case.
Gaussian Hellinger distance (GH). When D, = KL, A(t) = t and p
2, then one has
D((u,a) a),
br, = a² +22 2abe 1l- This cone distance (Burago et al. 2001 is further gener-
alized by De Ponti 2019) who shows that D is distance for power entropies (p(s) 5" p(s 1) -1
p(p 1)
pf 2 1 (the case = corresponding to D, = KL).
distance (WFR). When D. KT. X(t) - log (tA and q = p = 2, then one has D[u,, al, [u, b1)² = a² + 62 2ab cos( + A lu- vl). This construction, which might seem peculiar, corresponds to the one used to make unbalanced OT a geodesic distance, as detailed in (Liero al. 2015) Chizat et al. 2018c).
Hellinger-Kantorovich (HK)
Peasserstein-Fisher-ee
Partial optimal transport distance (PT). When D. TV, A(t) t?, q 2 and p = then D([u, a), [v, 6))* = + b (a A (2 - |as u defines a cone distance (Chizat et al. 2018c).
2.2.2 DEFINITIONS AND PROPERTIES
The conic formulation consists in solving a GW problem on the cone, with the addition of two linear constraints. Informally speaking, Le from Lemma becomes D, the term (I(p 4)= [( @ +) is taken into account by the constraints (5) below, and the variables (f.g) are replaced by (r² sP)
reads CGW(X,)) def infoeu, H(a) where
H((a) def. / (((xx(2, (2,, x'), [dy (y. s/) ss (d')), r), [v, dla 1' s'D),
Up v) def. ( a (QEM+(((X) C(D) f Pdaa (.r) = M. JR+
(5)
pdaaz(, =u]
It is similar to the conic formulation of UW, see Appendix A lation 22 and in sharp contrast with the conic formulation UW - here the transport plans are defined the cone { x C[Y] but the cost D is distance [[R+].
Note that similarly to the GW formu-
We present now the main contributions this paper, proved in Appendix C We state that CGW defines distance under conditions that hold for the settings of Section 2.2.1 and that it is upper- bounded by UGW, While the distance C/ cannot be casted as finite dimensional program even in discrete settings (because it is defined on a lifted space), UGW can be approximated with efficient numerical schemes as detailed in Section [3] The tightness of the bound between UGW and CGW and the computation of CGW are open questions left for future works.
Theorem 1. (i) The divergence CGW is symmetric, positive and definite up to isometries. (ii) Df is distance on +[ then CGW is a distance on the set of mm-spaces up to isometries. (iii) For any (De, >. p. q) with associated cost D on the cone, one has UGW > CGW.
Sketch of proof (i) CGW is positive and symmetric. Definiteness holds thanks to Proposition 3 (ii) The triangle inequality is similar to balanced OT, and applies the gluing lemma (Villani 2003
Lemma 7.6). The non-trivial part is showing that the latter lemma holds, because it glues two plans provided they have a common marginal. Since CGW is invariant under radial rescalings (called dilations in Appendix C). it is possible to dilate plans such that they have common marginal
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and remain optimal. (iii) Take optimal plan for UGW(X,)), From this one can build plan a such that ((T) 2 H(a) because L. 2 He- Furthermore a E Up. and is thus admissible and suboptimal, which yields UGW(X,)) (TT) H(a) 2 CG)().))).
ALGORITHMS
The computation of the distance CGW is in practice out-of-reach because it requires optimiza- tion over a lifted conic space, which would need to be discretized. We focus in this section on the numerical computation of the upper bound UGW, using an alternate minimization coupled with entropic regularization. The algorithm is presented on arbitrary measures, the special case of dis- crete measures being particular case. The discretized formulas and algorithms detailed in Appendix see also Chizat et al. 82)1 Peyré et al. (2016). All implementations are available at https: tttp://itttbbtat thub. com/ anonymous
cwonffrrence submi ssi
In order to derive a simple numerical approximation scheme, following Mémoli (011), we intro- duce lower bound obtained by introducing two transportation plans. To further accelerate the method and enable GPU-friendly iterations, similarly to Gold et al. (1996;; Solomon et al. 2016). we consider an entropic regularization. reads, for any =20
UGW
def. inf L(T) + &eL @ (T| @ v) 2 inf F( + eKI(T @ 1010 8 0) s)82)
(6)
a,7
and F(T,,) x2
X(Ixx ly 88 + (T( s nlle @ n)) + Do (x2 00 22/ @ v),
where (71.2g) denote the marginals of the plan y. Note that in contrast to the entropic regularization of GW Peyré al. (2016), here we use a tensorized entropy maintain the overall homogeneity of the energy. A simple method to approximate this lower bound is to perform an altemate minimiza- tion on ı and 7, which known to converge for smooth oo to a stationary point since the coupling term in the functional is smooth (Tssng 2001).
Note that if π @ y optimal then so is (ST) @ (17) with 2 Thus without loss of generality we optimize under the constraint m(r) =m(r) by setting s = /m((r)/m
In general, this bound not expected be tight, but empirically, alternate minimization often converges a solution with =y (as already observed for instance in Ranggaajann et al.] (1999); [Solomon et al. (2016), so that the algorithm also finds a local minimizer of the UGW; problem. In the Balanced-GW case in Euclidean spaces, the optimum is known to satisfy π = 7 (Konno 1976) and alternate descent is equivalent to a mirror descent algorithm (Solomon et al. 2016). Minimizing the lower bound of (6) with respect to either π y non-trivial for an arbitrary sp. We restrict our attention to the Kullback- Lebblck-Leiiler Libler case D2 = pKT, with >>> O, which can be addressed by solving a regularized and convex unbalanced problem studied in Chizat et al. (0188) Séjourné al. (201)). It is explained in the following proposition.
Proposition For a fixed 7,, the optimal æ E arg min F(x, 7) + <KT.(T & 71( @ v) is the solution of min [c, (x, y)d (2, v) + ( + pm)(t)
+ em(r) (T|H @ v),
where m(7) del. 7(X x Y) is the mass of 7, and where we define the cost associated to y as du )dy + p log( dra
(x, drt [((dx(E,)) dy (y. DDd + p log( dr
di )d + log( dr dadu )dy
Computing the cost c for spaces and Y of n points has in general a cost On in time and memory. However, as explained for instance in Peyré et al. (016)). for the special case X(t) = t? this cost is reduced to (1²²)) in time and (n²)) ) in memory. This is the setting we consider in the numerical simulations. This makes the method applicable for scales of the order of 10t points. For
larger datasets one should use approximation schemes such as hierarchical approaches (Xu et al.)
2019) or Nyström compression of the kernel (Altschuler et al. 2018
The resulting alternate minimization method is detailed in Algorithm I see Appendix D for a dis- cretized version. It uses the unbalanced Sinkhorn algorithm of of'Cai et al. (2018a); Séjourné et al. (2019) as sub-iterations and is initialized using μ @ v v/ (m(())m(())) This Sinkhom algorithm operates over a pair of continuous functions (so-called Kantorovitch potentials) f(x and g(y). For discrete spaces X and Y of size n, these functions are stored in vectors of size n,, and that integral

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Algorithm 1 UGW(X, 3/, 2,
Input: mm-spaces (X'))) relaxation p. regularization
Output: approximation (TT, 7) minimizing 6
1: Initialize π 7
@ ht///p///ii///eo// mm(z)m(p).g=00
2: while (T, has not converged do
3: Update π < <-,, then - Ct Ā m(<)p,, t m(x)E
4: while (f.g) has not converged do
5: Var, f(x + z4p
EΦ
log
6: Vy, gly) - ビ+カ
Eá log
(feldli)dd= du(())))))))))))))))))))))))))))
7: Update ((x, v) t- exp (J(x) + +(()) cz, w)) ((x))())
8: Rescale 7 + vm(T)/m(7))
9:
Return (T,
involved in the updates becomes sum. Each iteration of Sinkhorn thus has cost n²², and all the in- volved operation can be efficiently mapped to parallelizable GPU routines as detailed in Chizat al. 20188)) Séjourné et al. (201)). Another advantage of using an unbalanced Sinkhorn algorithm is its complexity O(n"/e to compute an E-approximation, as stated in Pham et al. (2020). which should be compared to O( /e"*)"") operations for balanced Sinkhorn.
Note also that balanced GW recovered as special case when setting p - +00, so that p/( + ū) + should be used in the iterations. In order to speed up Sinkhorn inner-loops, especially for small values of E. one can use linear extrapolation (Thibault et al. 2017) or non-linear Anderson acceleration (Scieur et al. 2016).
There is an extra scaling step after computing y involving the mass m(r). It corresponds to the scaling of Y such that m(T) m(y), and we observe that this scaling is key not only to impose this mass equality but also to stabilize the algorithm. Otherwise we observed that m(y) < 1 < m(T)
and underflows whenever m(7) 0 and m(x) - 00.
NUMERICAL EXPERIMENTS
This section presents numerical simulations on synthetic examples, to highlight the qualitative be- havior of UGW with respect to mass variation and outliers. In all these experiments, μ and are probability distributions, which allows us to compare GW with UGW.
Robustness to imbalanced classes. In this first example, take X = Y R²2 and consider E,C and S to be uniform distributions on an ellipse, a disk and a square. Figure [ contrasts the transportation plan obtained by GW and UGW for a fixed μ 0.5E + 0.5C and obtained using two different mixtures of & and S. The black segments show the largest entries of the transportation matrix n, for sub-sampled ing induced by the plan. Furthermore. the width of the dots are scaled according to the mass of the marginals π1 a and T22 1/, i.e. the smaller the point, the smaller is the amount of transported mass. This figure shows that the exact conservation of mass imposed by GW leads to a poor geomet- rical matching of the shapes which have different global mass. As this should be expected, UGW recovers coherent matchings. We suspect the alternate minimization algorithm was able to find the global minimum in these cases.
of points (to ease visibility), thus effectively displaying the match-
Influence of and debiasing. This figure (and the following ones) does not show the influence of E. This parameter is set of low value = = 10 2 on a domain [0, 1]2 so as to approximate the optimal plan of the unregularized UGW problem The impact of similar to those of classical OT, namely that it introduces an extra diffusion bias.
Robustness to outlier convention Figure 1 It corresponds to the moons dataset with additional outliers (displayed
Figure shows another experiment on a 2-D dataset, using the same display
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GW
UGW
GW
UGW
Figure 1: GW vs. UGW transportation plan, using 0.35 + 0.778 on the left, and 0.7E on the right.
GW
UGW, P 10
UGW P 10
UW.p=10- 10-2
Figure 2: GW and UGW applied to two moons with outliers.
in cyan). Decreasing the value of (thus allowing for more mass creation/destructioon in place of transportation) is able to reduce and even remove the influence of the outliers, as expected. Further- more, using small values of tends to favor "local structures", which is a behavior quite different from UW (I. Indeed, for UW, - O sets to zero all the mass of æ outside the diagonal (points transported), while for UGW, it is rather pairs of points with dissimilar pairwise distances which cannot be transported together.
Graph matching and comparison with Partial-GW. We now consider two graphs (X, Y) equipped with their respective geodesic distances. These graphs correspond to points embedded in R and the length of the edges corresponds their Euclidean length. These two synthetic graphs are close to be isometric, but differ by addition or modification of small sub-structures. The colors e(x) are defined on the "source" graph X and are mapped by an optimal plan ı on y E Y color [.] Jx c(x))x (I. y). This allows to visualize the matching induced by GW and UGW for varying P. displayed in Figure The graphs for GW should be taken reference since there is no mass creation. The POT library (Flamary & Courty) 2017 is used to compute GW.
For large values of P. UGW behaves similarly to GW, thus producing irregular matchings which do not preserve the overall geometry of the shapes. In sharp contrast, for smaller values of p (e.g. p 10-1), some fine scale structures (such as the target's small circle) discarded. and UGW is able to produce a meaningful partial matching of the graphs. For intermediate values (p = 10"), we observe that the two branches and the blue cluster of the source correctly matched the target, while for GW the blue points are scattered because of the marginal constraint.
Figure4shooe shows a comparison with Partial-GW (Chapel et al. 2020), computed using the POT library. It is close to UGW with a TV penalty, since partial OT is equivalent to the use of a TV relaxation of the marginal. UGW with KI, penalty first computed for a given P. then the total mass m of the optimal plan is computed, and is used as parameter for PGW which imposes this total mass as a constraint. Figure 3 and 4 display the transportation strategy associated to both methods. KL- UGW operates smooth transitions between transportation and creation of mass, while PGW either performs pure transportation or pure destruction/crration of mass. This can be observed Figure where nodes of the graphs removed and not taken into account by the matching. Note also that since PGW is equivalent to solving GW on sub-grap aab-graphs, the color distribution GW and PGW are the same.
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bbb.
Source X
Target Y
p= 0.1
p=1
p 10
GW (p=
Figure 3: Comparison of UGW and GW for graph matching.
C
c 0000
Source X
Target Y
m=0.64
m=0.93
m=
GW (m=1)
Figure 4: Comparison of Partial-GW for graph matching. Here is the budget of transported mass.
5 CONCLUSION AND PERSPECTIVES
This paper defines Unbalanced Coomov-Wasserstenn formulations. We prove that they are both positive and definite. We provide a scalable, GPU-friendly algorithm to compute one of them, and show that the other is distance between mm-spaces up isometry. These divergences and distances allow for the first time to blend in scamless way the transportation geometry of GW with creation and destruction of mass. This hybridization the key to unlock both theoretical and practical issues. This work opens new questions for futures works. On the theoretical side, the geodesic structures induced by unbalanced GW distances and divergences is an important subject of study. On the practical side, removing the bias introduced by the use of entropic regularization is important for applications to ML. Note that such a debiasing was successfully applied for Balanced-GWV in Bunne et al. (2019) and is shown to lead to valid divergence for balanced OT in Feydy et al. (2019b) and OW in Séjouréé et al. (0019). The design of efficient numerical solvers for the the conic formulation also interesting avenue for future works.

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A BACKGROUND ON UNBALANCED OPTIMAL TRANSPORT
Following Liero et al. (2015), this section reviews and generalizes the homogeneous and conic formulations of unbalanced optimal transport. These three formulations equal in the convex setting of UOT. Our relaxed divergence UGW and conic distance CGW defined Section build upon those constructions but are not anymore equal due to the non-convexity of GW problems.
A.1 HOMOGENEOUS FORMULATION
To ease the description of the homogencous formulation, we develop and refactor the Csiszär diver- gence terms of ① in form analog to Lemma It reads
/ Liid(a (x) (())))(()))(() v) + vho (1101 |
(W 1)a
inf =EM(X2)
where s) def, c rp(1/r) + st((/s), 1p def p (X) and (f del densities of the Lebesgue decomposition of (H, v/) with respect to (TT, π2)
dm 9
du
def.
d
are the
and
== = x= and == ga2 + vt.
Such form is helpful to explicit the terms of pure mass creation//dstruationn (1++1 + ( and rein- terpret the integral under и as a transport term with a new cost Lx
Then the authors of Liero et al. (2015) define the homogeneous formulations HUW
[ Hxdd (f())) ((v)) v) + (1++t |=-),
HUW(H, del inf TEM(X²)
(9)
where the 1-homogeneous function He is the perspective transform of T.c
Hc(r,s)
del.
inf ((c + (b)) + (())) 820
inf 6L. (T= (). 820
(10)
By definition one has Le 2 He, though after optimization one has UW = HUW.
A.2 CONE SETS, CONE DISTANCES AND EXPLICIT SETTINGS
The conic formulation detailed in Section A.3 is obtained by performing the optimal transport on the cone set C( particle. Coordinates of the form (ar, 0) are merged into single point called the apex of the cone, noted ox. In the sequel, points of X x R are noted (I, and those of C[X] are noted [ær r] to emphasize the quotient operation at the apex.
x IR, / x {()), where the extra coordinate accounts for the mass of the
For pair (p: q) E IR.,, we define for any [xr, r). [V. 8) E E[x]²
Dexxi ([x,, r], [y, s)) def HAid(awwi (r", sP).
(11)
In general Dejx] is not distance, but it is always definite as proved by the following result. Proposition 5. Assume that is definite, x-1(1)) = (0) and y (-1((-0))
{1). Assume also that for any (r, s), there always exists 0* such that Hr) = 0" T.o 3. Then Dexx] is definite on 0)
C[X}, i.e. Deix](1x, r), (y. a)) 0 if and only if (r
and x y).
Proof. Assume Decx (Ix, r], [v. s)) O, and write 0* such that Dejx]([x, r], s))a 0" (p )²)
0* X(d(x,)) + r p()( + spp("))
where the last line is given by the definition of reverse entropy. There are two cases. If 0" > 0.
since all terms are positive, there are all equal 0. By definiteness of d it yields x = and
because ({0)) (1} we have pP 3d 0 and r == s. If 0* then De ([x, r], (y, s))s
(rP The assumption i '((0)) {1) implies (00) 0, thus necessarily
= 0. O
The function H. can be computed closed form for certain number of common entropies 4o and we refer Liero et al.] (2015) Section 5) for an overview. Of particular interest are those yp where Deix] is a distance, which necessitates a careful choice of X, and We now detail three particular settings where this the case. In each setting we provide (D,,, X, P. g) and its associated cone distance De[x]-
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Gaussian Hellinger distance It corresponds
D2 = KL,
A(t) = ? and = = 2,
Dex ((2, r]. 8)) = 5² 2rse d(z.u)/2
in which case it is proved in Liero et al. (0155) that De[x] is cone distance.
Dellingr-Kaantrrviic Wasserstein-Fishen Riso distance It reads
D KI, X(1) -- log cos 2 (LA 5) and q --p 2, (1 r], (v, a))²
8² 2rs cos( 3 A d(x, u));
in which case it is proved in Burago al. (2001) that De[x] is distance.
The weight X()
log ( 5), which might seem more peculiar, is in fact the penalty that makes unbalanced OT a length space induced by the Gaulsian-Helllnngrr distance (if the ground metric d is itself geodesic), as proved in Liero et al. (2016); Chizat et al. (2018b). This weight introduces cut-off, because A(d(z, y)) = too if d( y) T/2. There is no transport between points too far from each other. The choice of / is arbitrary, and can be modified by scaling À + (-/s) for some cutoff s.
Partial optimal transport corresponds to
and q and p 1, Desxy ([x,, rl, [y, s))* - (r s) - d(x, y)
D2 = TV, TV, A(t) = t
which case it is proved in Chizat al. (2018c) that De(x] is a cone distance. The case Dj = TV equivalent to partial unbalanced OT, which produces discontinuities (because of the non non-smoothness of the divergence) between regions of the supports which are being transported and regions where al. (2015) do not mention that this Detx; defines
mass is being destroyed/created. Note that a distance, so this result is new to the besi of our knowledge, although it can be proved without a conic lifting that partial OT defines distance as explained in Chizat al, (2018c).
A.3 CONIC FORMULATION OF UW
The last formulation reinterprets UW an OT problem on the cone, with the addition of two linear constraints. Informally speaking, He becomes De[x]. the term (1x+1] + (u-1) () taken into account by the constraints below, and the variables (f.g) are replaced by (r", sP). It reads
/ (L,))(
CUW(H,
inf aEll,(u.)
r), [v sl),
(12)
where the constraint set Up v) is defined
/ r'da1(.7) ui / s'daz(+,))=v)
Up(p, def. a M+
(13)
Thus CUW consists in minimizing the Wasserstein distance WD (an, 02) the cone (e(x),eex)) The additional constraints on (21,02) mean that the lift of the mass on the cone must be consistent with the total mass of (H,z). When De(x] is distance, CUW inherits the metric properties W Our theoretical results rely on an analog construction for GW.
The following proposition states the equality of the three formulations and recapitulates its main properties. The proofs are detailed in Liero et al. (2015).
Proposition (From Liero et al. (2015)). One has UW = HUW = CUW, which are symmet- ric. positive and definite. Furthermore, if (X, dx) and (e(x),De(xi) metric spaces with X separable, then (XX endowed with CUW is a metric space.
Proof. The equality UW = HUW is given by Liero et al. 2013 Theorem 5.8), while the equality HUW CUW holds thanks to Liero al. 2005 Theorem 6.7 and Remark 7.5), where the latter
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theorem can be straightforwarly generalized to any cone distance built as in Section 2.2.1 Since De(x] is symmetric, positive and definite (see Proposition 3), then so CUW. Furthermore, Deix] satisfies the triangle inequality, separability of X allows to apply the gluing lemma (Liero et al. 2015 Corollary 7. 7.14) which generalizes to any exponent p defining vp(u,v) and any cone distance De(x)-
B UGW FORMULATION AND DEFINITENESS
We present in this section the proofs of the properties of our divergence UGW. We refer to Sec- tion for the definition of the UGW formulation and its related concepts. For conciseness we write [(x, 1', 3/) = (xxxxxxx) dy (y, y/)l-
We first start with the existence of minimizers stated in Proposition ! It illustrates in some sense that our divergence is well-defined.
Proposition 7 (Existence of minimizers). Assume (X,3) to be compact mm-spaces and that we either have
1. ip superlinear, i.e sp
2. X has compact sublevel sets in R4 and 24 inf >>>
Then there exists E MM4(X x Y) such that UGW = C(x).
Proof. We adapt here from Liero al. 2015 Theorem 3.3). The functional is lower semi- continuous as a sum of l.s.c terms. Thus it suffices to have relative compactness of the set of min- imizers. Under either of the assumptions, coercivity of the functional holds thanks to Jensen's 's inequality
[(T) 2 m(T)² inf A(T) + m(n) m(T)²
kp( m(xx²
+ m(r)2pp
m(u)²
m(v)²
m(n)² m(T)²
m(v)²
m m(T)²
2 m(T)² ( iff A(T) (TT) + m(T)² m(u)²
m(T)²
m(v)2
As m(T) - +oo the right hand side converges to 24' inf X > O, which under either one of the assumptions yields [(T)) + +oo, hence the coercivity. Thus we can assume there exists some such that m(T) < M. Since the spaces assumed to be compact, the Banach-Alaogla theorem holds and gives relative compactness in M+ (X x Y).
Take any sequence of plans тr that approaches UGW(X,)) = inf C(T). Compactness gives that a subsequence Tne weak* converges to some πτ. Because L l.s.c, we have L(x*) < inf (TT), thus [(x*) inf L(r). The existence of such limit reaching the infimum gives the existence of minimizer.
Note that this formulation is nonegative and symmetric because the functional L is also nonega- tive and symmetric in its inputs (X' Y).. This formulation allows straightforwardly prove the definiteness of UGW.
Proposition 8 (Definiteness of UGW). Assume that ( = {1} and X x- = {0). The following assertions are equivalent:
1. UGW(X,)) 0
2. 3R M4 Y) whose marginals are (p,v) such that dx (2, dy(y, for
a.e. (2,x² y') E (X x 2)2.
3. There exists mm-space (Z, dz. m) with full support and Borel maps 4/x 7 - X and 4/ 7 > Y. such that (wx ) 1s, (by)): = v and dz (ux) = (1>y Pdy
4. There exists a Borel measurable bijection between the measures' supports & : spt(H) + (v) with Borel measurable inverse such that 1/ypu and dy dx
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Proof. Recall that (2) (3) (1) + (2).
from Sturm 2012 Lemma 1.10). thus it remains to prove
If there is such coupling plan between (H, then one has @ T-a.c. that I 0, and all (p- divergences are zero as well, yielding a distance zero a.c.
Assume that UGW (X, )) 0, and write under our assumptions we have = O, m1 @ #1 = f @ μl and T2 @ T2 = V @ v. Thus we get that g has marginals (J,, 2/) and that dx (z, x²) dy (y, y --a.e
optimal plan. All terms of are positive, thus
We end with a result on the reformulation of UGW which is the first step to connnect it with the conic formulation CGW.
Lemma 2. Defining Le(r, s) def + r1((1/) sip(1/s), and writing (f del du
del de the
drs .9 dmg
Lebesgue densities of (je, v) w.r.f. (T), such that μs f + x- and v = g + vt, one has
[(T) - xxaxY2
La f. g)dddd =
Proof. Using Equation
23 one has
[(x)
X(T) + DD (T)lI))) + +D
X2XY=
1xexxa X(T)ddd + D (m|m)) D
-
vff & fd f ( @ g)dzzdm2
XxXXY= X(T)ddd +
+ +000)((( @ (1) (()
(a & f.g 8 gdddd x 11
X2xx2
C CONIC FORMULATION AND METRIC PROPERTIES
We present in this section the proofs of the properties mentioned in Section 2 We refer to Section and Appendix for the definition of the conic formulation and its related concepts.
In this section frequently use the notion of marginal for neasures. For any TT(E) E x F + E the canonical projection such that for any (x, y) E E x F, (3)) y) X. Consider two complete separable mm-spaces
F, F, write
= (X, n4) and ) = dy v). Write π E M+ Y)= coupling plan, and define its marginals by definition of the marginals can also be seen by the use of test functions. In the case of reads for any test function g
T and n2 T ()
n. The
/ ((z)d<]()) = = ((x)dT(Z.)) v).
C.I PRELIMINARY RESULTS
We present in this section concepts and properties which are necessary for the proof of Theorem We introduce dilation operator whose role is to rescale the radial coordinate of a measure with given scaling.
I
Definition 2 (dilations). Consider v( r), [y, s)) a Borel measurable scaling function depending on [z, r), []. E C|X] x C(Y]. Take plan E (+ x e(Yl). We define the dilation Dil, a (ho) where
ho( s)) del ([x, r/o, w. s/w)),
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where o([r, r), s)). It reads for any test test function Ç
/ E((z, r), ( sl)dDil, (a) = 5((x, ly s/ s/wl)w"da.
The importance of dilations is given by the following lemma. Lemma 3 (Invariance to dilation). The problem CGW is invariant to dilations, i.e. for any a e Up(p,)) v), we have (a) Up(,,v) and H(a) H(Dil.(a)).
Proof. First we prove the stability of Up(p, v) under dilations. Take a Up(u, For any test function & defined on X we have
J
- p( = ((x)r" ((z)dp(Z).
Similarly we get p.
(a)) v, thus (a) E Up((,, v).
remains prove the invariance of the functional. Recall that D9 is p-homogeneous. It yields
H(Dil.(a))
D(Idx(1, r'r, ld (,, (idd
/ D(Idx(z, x') _ = Idr S =)) v dada
/ u2p D((dx (z, x)) rr', (dy (v. v)) ss'))) 2 doda
= / D([dx (z. x rr (dy (v. v/), ss dada
-(aa)
Both the functional and the constraint set are invariant, thus the whole CGW problem is invariant to dilations.
The above lemma allows to normalize the plan such that one of its marginal is fixed to some value. Fixing marginal allows to generalize the gluing lemma which is key ingredient of the triangle inequality in optimal transport.
Lemma 4 (Normalization lemma). Assume there exists a E Ug(u, v) such that CGW(X, y) = H(aa). Then there exists â such that Vp(H,,) and CGW(X,)) H(ā) and whose marginal on C ve[y] = (p(e(r))á = -oy + Pp @ 81), where P is the canonical injection from Y x R2 C(Y).
Proof. The proof is exactly the same as Liero et al. 2015 Lemma 7.10) and is included for com- pleteness. Take an optimal plan Because the functional and the constraints are homogeneous in (r, s). the plan à = a + dox & doy verifies E (p, v) and H(a) = H(a)). Indeed, because of this homogeneity the contribution dox & doy has (r, (0,0) 0) which has thus no impact.
Considering à instead of a allows to assume without loss of generality that the transport plan charges the apex, i.e. setting
S ([x, lu. E C[X]
CYy, [v. s) =or
(14)
one has wy def. á(S) 2 1. Then we can define the following scaling
s if > 0 wy 1/9 otherwise.
o[ s),
(15)
We prove now that (å) has the desired marginal on C(Y) by considering test functions {(p,. s))- We separate the integral into parts with the S, and write â ãls + alse their restrictions
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to S and S respectively. It reads
I E(vv ())
/ E(lv, s/vl)odda
/ E(Vv, s/bl)odd alls + / s/u)o// als-
/ E(or juy d alls / s/lv.//s))8 als-
E(oy) wy wp + / Ellv. dā
E(or) + / E(p(v, ())(((()) 8 61(s))
/ <(Iv s))d(oo + p=(
which is the formula of the desired marginal on C[r]. Since á E Up(H,,)) its dilation is also in Up(H,)) v),, and H((a) = H(a) (a)i..
C.1.1 PROOF OF THEOREM 1
Non-negativity and symmetry hold since H. is sum of non-negative symmetric terms. To prove Definiteness, assume CGW(X.)) = 0, and write a an optimal plan. We have @ a-a.e. that xx(x,2') = dy (y. y/) and rr' ss' because completeness of (X and result from (Strr (2012) Lemma 1.10), such property implies the existence of a Borel isometric bijection with Borel inverse between the supports of the measures di Supp(u) + Supp(v), where Supp denotes the support. The bijection & verifies dx (x,xx)) dy (vxz), (z))). To prove X' y it remains to prove dyu = v. Due to the density of continuous functions of the form ((x))(x'), the constraints of Up (1,, (1,,V) v) are equivalent to
is definite (see Proposition 3). Thanks to the
/ (mr")dai(ss))dd = M S..
[ (edd"da((s))daa((
VV..
Take a continuous test function & defined on Supp(v)?. Writing y (xx) and D' = (bxx), one has
f Ely. v' ydvdv
/ Ely, (ssss'ssaaa
/ ((2))))
/ E(v(x), 0b ((r') dada
/ E(v(x)), ((aa))))))))))))) dpdd
/ É( x' (ddddddd/dd/ee
Since 1 is a bijection, there is bijection between continuous functions & of Supp(v)² and functions Ć of Supp(u)². Thus we obtain v = wiH and we have X' ~ y.
It remains to prove the triangle inequality. Assume (X, y, Z) respectively equipped with measures (14, 1/, 17). consider a, 3 which are optimal plans for CGW(X,)) and CGW()), 2). Using Lemma 4 to both and 3, we consider measures (ā, 3) which are also optimal and have a common marginal i on C[Y] Thanks to this common marginal and the separability of (X, Y, Z), the standard gluing lemma (Villani 2003 Lemma 7.6) applies and yields glued plan 7 E MM x C[Y) c(Z]) whose respective marginals on C[X] x C(Y] and C[Y] x C[Z] are (ā, 3). Furthermore, the marginal & of y on C[X] x C|Z] is in U4 (u, n). Indeed,
that D satisfies it. Given three mm-spaces
(7, a) have the same marginal [[X] and same for (5: (3) C[Z], hence this property. Write
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Under review as a conference paper at ICLR 2021
dx xx,), for sake of conciseness (and similarly for Y, Z). The calculation reads CGW(X,
(16)
< ([ D(idx,rr rr'), "r,,,,d Idz, ((r,, [2, (2d (rl,,,
(17)
( / D((dx. Irz, [dz, ( Iv ())d.t)ddlllll W. w'..s...... (.,"")))))))
(18)
/ (D(ldx (sy, D( s8'1, [dz, 14 1)))
(19)
$ / D(ldx, mr) ldy, ss Iyyydd)) 9 + [fDiddy ss [dz. (t1))d7d7))
(20)
/ p(ldx, rr'), [dy, se Io')) r), Iv, sl)daa((a', r'). M/.. o/))
+ ([ D(dddr Idz,,
[dz, ut 4t)) s), ()ddd//wwwwwww "oddB ,) "
(21)
I CGW (X, VY + CGW()), 2)
(22)
Since 7 E Up(u, n). it is thus suboptimal, which yields Equation 17 Because fy is the marginal
of y we get Equation (88). Equations (19) and (20) are respectively obtained by the triangle and Minkowski inequalities, which hold because which is a distance, Equation (2T) is the marginal-
ization of 7, and Equation (22) is given by the optimality of (ā, B), which ends the proof of the triangle inequality.
C.1.2 PROOF OF THE INEQUALITY BETWEEN UGW AND CGW
The proof consists in considering an optimal plan for UGW, building a lift of this plan into the cone such that [(T)= 2 H((a), and prove that a is admissible for the program CGW, thus suboptimal.
Using Equation
8 we have
@ μ (f & f (m () /² o (fm) + opt
(23)
V v 8 (g
8) د 2 + 8 Or
Recall that the canonic injection reads p(x, r) = [x. r]. Based on the above Lebesgue decomposi- tion, we define the conic plan
a = (p(z, f(x) ), ply. (()) )): (z, y) + 8ox @ pp[p- X 81] + palu- @ 61] 2 doy
(24)
We have that a E (g v). Indeed for the first marginal (and similarly for the second) we have for any test function { (cr)
f {())())daa (())s() + 0+ ((=)(())a))+())
f ((x)d((()) +
((x)dp(x).
We define 0+ = (r,ss the parameter which verifies Ho(r, 8) = 0" Lo s/0*). We restrict to the set S (rr ((rr'),, (ss')P) > 0}. By construction, (r, is 1-homogeneous in (r, s). Thus on S we necessarily have r,r', 2,r',,, 8 > O. It yields
als (p(2, f())) g(v) F). p(a', f(d)) p( g(0')) (())))())))))(((())))))))))))))) ())) x).
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Under review as a conference paper at ICLR 2021
Concerning the orthogonal part of the decomposition, note that whenever 0 = 0, due to the defini- tion of the cone distance reads
D[ r], (y. s))² = =(()(r² +000p++* *²).
(25)
geometrically means that the shortest path between [x,, r] and [v, s) must pass via the apex, which corresponds to pure mass creation/destrrutiion regime.
Furthermore we have that
((4 2 1011
[1 (( 1 (f
Indeed, thanks to Equation 24 we have for the first marginal that
(1u 2 u) = (ut @ (fTI) + (fTi) @ pt + at @ (x3)
/ (rr')ddp:lut @ ddjdd((z', f(x') )TT) (z')
+ / (r))) (x) oú]
/(r')Pappllt @ d 06
= (
Note that the last equality holds because each term of @ a involving measure 6ox cancels out when integrated against (rr')?.
Eventually the computation gives (thanks to Lemma D
xxxxx J La @ f.g g) +
((T)
/ (ann
2 / pl f000 (a @a)ls.
2 /
lol)e 8))s
+ʃ x(0)((T)))) + (sd))) (a aa)ss
2 / (11x(2,z'), (y. v'), ss ss
2 H(a).
Thus we have UGW(X,)) = L(T)) 2 H(a) CGW(X,)).
D ALGORITHMIC DETAILS AND FORMULAS
D.1 DECOMPOSITION OF KL QUADRATIC DIVERGENCE
We present in this section additional property the quadratic-KL divergence which allows to reduce the computational burden to evaluate it by involving the computation of a standard KL divergence.
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Under review as a conference paper at ICLR 2021
Proposition 9. For any measures (H,)2) (2))), has
KT,p @ 1 @ [) m())KI((uaa) + m(N)KL(v13)
(26)
+ (m((u) - (m(a))(((() (3)).
In particular,
KI(W & /// = 2mm())KT((Ap) + (m(u)) m(v))²
(27)
Proof. Assuming K (3 be finite, one has μa= fa and 99.. It reads
KL(u @va @ B) / log(t @ g)dudd m())m(()) m
= =m(v) / log(f)d - mlH) / log(g)de
m(u)m(v) m(()m((3)
m(4) (K.((a)))))))) +
m(a)
(m
= ([())L((la)++(()((((
(mm())(())-mmm(((( m(u)m(B) ((( m(a)m(B)
= + (
+(m(p) ((())(((()) m3)))
We now prove Proposition 4 which applies the above result. Proposition 10. For a fixed 7, the optimal æ E arg min F(T, 7) + sKL(T @ v @ w)82) the solution of min J c5 (Ir, y)dr(x v) + pm
+ where ml()) 7(X x Y) is the total mass y, and where we define the cost and weight associated to t as
& v),
(x,1) del.
(ATT(a,..)))) + p / log( dzl dnn Jdn du
p / log(
dra dr +
dy log(
ddy. dpdr
dv
Proof. First note that F(y, T) = F(T, 7) so that minimizing with the first or the second argument gives the same solution. The rest follows from the relation
KL(TI & nlp pu) (LL))L( + m()) (m) (m(y) m(H))(((TT) (m))))
and also from KL(T|PP) = f log( dp )dh (m(7) - m(n)). Similar formulas hold for ( 72) and 7).
D.2 DISCRETE SETTING AND FORMULAS
In order to implement those algorithms, one consider discrete mm-spaces X = (zi) and Y (",)7 1. endowed with discrete measures = E, 1502, and v = E, Ujoyz, where μ4 Vj 2 0. The My). Transport plan are thus also
distance matrices are D,,,, del (x( and DX,, discrete ग = Σi, (x..))))))))
The functional L now reads in this discrete setting
- (y. dT) ()
Σ(D
i..ke
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Under review as a conference paper at ICLR 2021
and KL(T ( TTIH = Σ
log
11, - Σ TI,ITIS + Eumo 4.
Hilli
2m ④T)
Σ
log
11,
=m(x)² + m(H)?
14
where we define the marginals 11.6 Σj Tk.j T2,8
E, Ti, and m(T) Ene Tt,j
= When one runs the stabilized implementation of Sinkhorn's iterations with a ground cost Ci. C(ri, Vj) between the points, it is necessary use Log-Sum-Exp reduction which reads
ft + EP where LSE, is reduction performed on the index j. It reads
E E+> LSE, [() Ct,s)/= = + log(H;)]
(28)
(Ct,(Ct.)) log
Σ exp(Cr. - max C(.a)) = max Ci
(29)
k
where the logarithm and exponential are pointwise operations.
Algorithm 2 UGW(XX p, p, in discrete form
Input: mm-spaces X' = (D (4:)) and y = (DY, (v5)g), relaxation regularization E
Output: approximation (Tr, 7) minimizing 6
Initialize matrix Tt, = %. u42u// M)) pz), vector 9j (x=0)
2: while æ has not converged do
3: Update π t-
4: Define m((T) + Diigien Miji s m(T) m(T)e
5: Define c <- ComputeCost(XX, y, n,, P.
6: while (f.9) has not converged do
7:
< gp log
Σ, exp ((9) - c2,) / log vj)
8:
9
Ep log
Σ exp ((fe c/))/E + log ui)
9: Update 74. < exp
(Js 9j cis)/ē] Mive
10: Rescale f- m /m/7)
11:
Return 7))
We also provide algorithm that computes the cost of defined in Proposition (O) We focus the case De = pKL and X(t) = t² which is computable with complexity Onn²) as shown in Peyré et al. (016). Indeed, note that one has
f (dx(2,,2) dy (2. 2'))''')7(<'''' (=('' dd / dr dr(v. (v. (2.d))d72(0)))
- 2 /
Algorithm 3- Computo V, n. 0, e) in discrete form
Input: mm-spaces X' = (D.,, (wi)s) and y (DY, K,8: (vy)g), transport matrix (TT relaxation
regularization e
Output: cost & defined in Proposition 10
1: Compute Æ1.j < Σe ji and 22 E, JzKk
m = and πिअ T
2: Compute A, - Σ (D PTss
> A = (D*)
3: Compute B₁ (n(D) T
B
4: Compute C., <- E, DXY
Ee DY Tj.k)
D C = D
5: Compute E p E, log
Tie
Tij + +P log
52.k V2
T222 + Ej.t log
pype Jj.kk
By
6: Return c c A, B 2C,tt E
22


Reviews:
[{"review_id": "ubzjMspIFIv", "reviewer": "AnonReviewer1", "summary": "", "questions": "", "limitations": "", "rating": 7, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "The authors consider the Gromov Wasserstein (GW) problem for metric measure spaces having different masses (i.e., Unbalanced GW). Similar to the ideas of unbalanced optimal transport (UOT), they proposed to use a quadratic divergence to relax the marginal constraints (instead of divergence as in UOT). When divergence is KL, the authors derive a GPU-friendly algorithm for the UGW  which relies on the unbalanced Sinkhorn algorithm. Additionally, the authors also propose a variant of UGW, namely Conic Gromov-Wasserstein (CGW) to address the different masses of metric measure spaces. The authors propose that CGW has nice properties (Theorem 1). However, there is no algorithm to solve the CGW yet.\n\n+ It is easy to follow the paper. The authors provide rigorous details for the unbalanced GW.\n\n+ For the UGW with divergence, it seems that it is not surprised for this idea by extending the ideas of entropic OT for unbalanced OT into entropic GW for unbalanced GW (e.g., using Sinkhorn iterations).\n\n+ The proposed conic GW is theoretically interesting since it comes up with no algorithm yet. The authors also draw a connection between CGW and UGW.\n\n+ It seems that the experiments are quite simple (with some toy examples). It seems better if the authors use UGW in some applications (e.g., some motivating applications for the unbalance case).\n\nSome of my other concerns are as follows:\n+ The authors propose to use quadratic divergence in UGW. Could the authors give more motivation/explanation about this approach, why not just divergence as in unbalanced OT (Although the GW is a quadratic assignment, the unbalanced problem for GW comes from the marginal constraints, why not just simply use the divergence between the marginal and measures)?\n+ The authors proved nice properties for the proposed CGW and draw its connection to UGW. However, CGW has no algorithm yet. I wonder whether there exist some special instances of UGW (with some specific divergence, e.g., total variation or KL), one may have some interesting properties as in CGW? \n\nOverall, I lean on the positive side. (i) The ideas of UGW may not be a surprise (together with its algorithm). (ii) The proposed CGW has good properties for GW problems for metric measure spaces having different masses, however, there is no efficient algorithm for it yet as (a trade-off). (iii) It seems better in case the authors improve experiments (e.g., use UGW in some motivating applications for the unbalanced case).\n", "extra_scores": {}, "extra_text": {}}, {"review_id": "qJFteJhbEv", "reviewer": "AnonReviewer2", "summary": "", "questions": "", "limitations": "", "rating": 6, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "The paper introduces a novel unbalanced Gromov-Wasserstein type problem. The Gromov-Wasserstein distance is very useful in practice for comparing probability distributions that do not lie in the same metric spaces. It has recently found several successful applications in ML for computational chemistry, graphs comparisons or NLP. Following previous works on unbalanced optimal transport (i.e. soft constraints over marginals enforcement of the coupling matrix), and the rationale that disposing of unbalanced versions of transport problems can alleviate in some ways presence of outliers or noise in the distributions, the authors propose two ‘unbalanced’ variants of the Gromov-Wasserstein (GW) problem, that allow comparison of metric spaces with arbitrary positive measures up to isometries (I.e. rigid transformations). \n\nThe paper is fairly well written, original, and the related works is particularly complete. The theoretical part of the paper is sound, rigorous and well-motivated, and I learnt many things from it. The idea of using a quadratic \\phi-divergence is neat and particularly clever. Yet, the paper requires, to some extent, very good notions in (unbalanced) optimal transport and probability, and I wonder, if some notations could have been eased a little bit (for instance, it seems that D_\\phi is usually chosen as KL), but I guess the choice was made to be as general as possible.  While I did not check in details all the proofs in the appendix Section, I believe the work is solid. The algorithm parts is a little less satisfactory, as it amounts to optimize a upper bound  of the described problem, following existing work from [Mémoli11]. Though appealing, this upper bound is known to lose some properties of the original GW distance (for instance, two close points in the source can be matched to two distant points in the target if they share a very similar ‘distance profile’ or distance distributions wrt. the other samples). It seems that this effect is observable in Figure 2, where parts of a moon are ‘flipped’ in the matching. Combined with entropy and unbalanced formulation, I guess the final result can be very hard to interpret in a practical setting. \nFinally, the weakest part of the paper is the experimental Section. Only two toy examples are presented. While the method could have been used in many settings (graph classification, embedding matching in NLP or  even graph matching, for which many algorithms exist, etc.), it is very hard to conclude about the practical interest of the method. \nWhile I guess this is not a problem if one focuses on the original contribution of this novel unbalanced distance, it is more problematic, in the reviewers perception, for a machine learning venue such as ICLR.\n\nFor this reason, I will solely suggest a ‘weak accept’ decision, while I really believe the theoretical sections have a lot of merits. \n\nMinor comments.\n\n- On page 1, it is said that « the paper defines for the first time a class of distances between the [aforementioned] objects ». This claim is a bit strong knowing that GW has already been used in several (cited) applications. I guess authors could be a little bit more precise on the meaning of this sentence;\n- Page 4, just after definition 1, it seems (at least with my PDF reader), that there is a problem in the symbol used for \\phi on the last line of the paragraph; \n- The claim that a cost of O(n^3) in time and O(n^2) in memory allows to scale to large problems is a bit strong. In practice, what is the maximum size of the problem that can be addressed in reasonable time with this method ? I expect that handling graphs that have more than 10k nodes for instance in very difficult. \n- The final rescaling of \\gamma (line 8 of Algorithm 1), is a little bit difficult to understand from the alternating minimization point of view. \n- In the experimental Section, which algorithm is used to compute GW ? Also, I guess Figure 3 could present the original graphs (without scaling the dots), to gain a better understanding of the original problem\n\n\nEDIT after rebuttal period\n---------------------------------------\nmany thanks to the authors for taking into consideration my comments. I decided not to change my note because I still believe the paper lacks of a significative experimental Section.  ", "extra_scores": {}, "extra_text": {}}, {"review_id": "aQTYzjP0clK", "reviewer": "AnonReviewer3", "summary": "", "questions": "", "limitations": "", "rating": 5, "confidence": 5, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "In the paper, the authors propose two versions of Gromov-Wasserstein distance when the weights of measures do not need to sum up to one. The first version, named Unbalanced Gromov-Wasserstein (UGW), is a direct application of unbalanced optimal transport (UOT) to the setting of Gromov-Wassertstein. The second version, named Conic Gromov-Wasserstein (CGW), is an extension of the conic formulation of UOT to the setting of Gromov-Wasserstein. The authors also show that CGW is a distance in the metric-measure spaces and is an lower bound of the UGW. Finally, the authors also provide some experiments with their proposed divergences.\n\nIn my opinion, the two proposed versions of Gromov-Wasserstein are quite interesting but lack novelty. They are simply extensions of the framework of UOT to the framework of Gromov-Wasserstein. Here are my other comments with the paper:\n\n(1) The CGW is introduced as a lower bound of UGW; however, it is computationally expensive and practically out-of-reach as the authors admitted in the paper. Therefore, I am a bit confused with the proposal of CGW in the paper as it is not useful in practice. The theory of CGW also does not appear deep enough for me to appreciate the contribution. Unless the authors can provide some thoughts on how to practically implement CGW in the rebuttal, I feel that the inclusion of CGW is quite weak and does not add extra practical value to the paper.\n\n(2) Going back to the UGW, it is a direct extension of the framework from UOT to that of Gromov-Wasserstein. The theories (Proposition 1,2 and Lemma 1) are fine and not hard to obtain given those from UOT. In the revision, I think the authors should give some examples of $\\varphi$ satisfying the assumptions in Propositions 1 and 2 to help the readers to understand these results better. In Lemma 1, the authors should provide some intuition about $L_{c}(a, b)$ as it is a key term used later to define the conic discrepancies $\\mathcal{D}$ in the definition of CGW.\n\n(3) There is an interesting algorithmic perspective of using unbalanced version over the balanced version. For the setting of OT, the recent paper of [1] shows that solving the UOT (by Sinkhorn algorithm) when the divergence is KL has the complexity of $n^2/ \\varepsilon$, which is near optimal, where $\\varepsilon$ denotes the desired accuracy to approximate the UOT by the entropic UOT. On the other hand, the complexity of using Sinkhorn algorithm for solving the OT is at the order of $n^2/ \\varepsilon^2$. Therefore, by considering the relaxation over constraints, solving UOT is in fact algorithmically favorable than that of solving OT. \n\nGoing back to the UGW, the relaxation of constraints, in my opinion, may also help the convergence of algorithm (specifically Algorithm 1 in the paper) comparing to that of the GW (via Sinkhorn algorithm) though the rigorous theoretical analyses of these algorithms can be very tricky in general. However, I think the authors also can provide some numerical experiments to see whether this phenomenon holds true. \n\n(4) The literature of using UOT as a robust version of OT in practical applications of deep generative models and domain adaptation has been considered recently; see for example the paper [2]. The authors may consider adding this reference in the related works and contributions.\n\n\nReferences:\n\n[1] K Pham, K Le, N Ho, T Pham, H Bui. On Unbalanced Optimal Transport: An Analysis of Sinkhorn Algorithm. ICML, 2020\n\n[2] Y. Balaji, R. Chellappa, S. Feizi. Robust Optimal Transport with Applications in Generative Modeling and Domain Adaptation. NeurIPS, 2020\n\n", "extra_scores": {}, "extra_text": {}}, {"review_id": "XGndPvBQ2ta", "reviewer": "AnonReviewer4", "summary": "", "questions": "", "limitations": "", "rating": 6, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "Summary and overall recommendation:\n\nThis paper introduces two generalizations of the Gromov-Wasserstein distance to the case where the measures are unbalanced (i.e., not necessarily probability measures). While one of these is a proper distance, it is computationally infeasible. The other one, an upper bound relaxation, is not a proper distance but is much more computationally feasible. The main contribution of the paper is to prove some fundamental properties of these objects: positivity and definiteness. The paper proposes an algorithm for solving the upper-bound version, and shows proof-of-point experiments on very simple 2D settings. \n\nDespite some shortcomings in novelty, experimental evaluation, and presentation detailed below, I find the theoretical contribution of this paper to be just enough to carry it, so I'm (weakly) recommending acceptance. \n\nStrengths:\n\n1. Albeit relying on ideas that are already quite popular in the OT literature (Gromov-Wasserstein distance, Unbalanced OT), this paper manages to weave them together in a very compelling way \n2. The paper has sound theoretical footing, and a makes a solid theoretical contribution.\n3. The writing and exposition are mostly clear, intuitive and well developed\n\nWeaknesses:\n1. The flip side of strength (1) is that the novelty of this paper is arguably limited, considering it builds on known techinques, and is addressing a problem that has been tackled in at least two recent works (De Ponti & Mondino 2020, Chapel et al. 2020).\n2. Some of the assumptions and results are not discussed in enough detail\n3. The experimental section is limited to very simple settings, that either conisder measures on the same space (defeating the purpose of GW) (Fig 1,2), or do so for very simple synthetic data (Fig3). In particular, none of the motivating applications mentioned in the introduction (NLP, chemistry, graph matching). \n\nOther comments:\n1. It would be useful if the paper discussed the assumptions that go into Prop 1 (superlinearity, compact level sets) in more detail. What families of $\\phi$-divergences satisfy these?\n2. I understand that Lemma 1 is a tool to draw a connection between the two approaches, but I find its introduction dry and abrupt. It would help to provide some discussion on the intuition on introducing such a perspective transform.\n3. The presentation of the conic formulation needs fmore hand-holding. What's the motivation for this cumbersome the conic formulation? Are there other prior examples of such conic distances (even if not on mm spaces)?\n4. I would have appreciated a discussion on the tightness of the bound of Theorem 1. Without this, the purported possibility to use one as a computationally feasible drop-in for the other is not fully supported. \n5. I don't think $\\gamma_1$ and $\\gamma_2$ in page 6 were formally defined anywhere\n6. There is no comparison to other unbalanced methods (either OT or GW) except for one setting in Fig 3. I suspect most of these alternatives would produce similar results in Fig 1, so what is the competitive advantage of this method? If it is complexity/runtime (vs other unbalanced GW methods), then there should be either a formal complexity analysis, or a empirical runtime comparison. Without these, the paper feels incomplete. \n\nQuestions:\n1. What is meant by the last sentence in the first paragraph? This reads to be as if this paper claims to propose the first distance between the objects tackled in those works, which is certainly not the case for all of those which use GW or variations of the Wasserstein distance. \n2. Is there a reference for the quadratic $\\phi$ divergences introduced in 2.1?\n3. Do any of the experiments use the debiased UGW mentioned in page 7?\n\nMinor comments and typos:\n* pg 3 shed some lights\n* missing \"is\" in first line of Prop 1\n* pg 8 \"close to be isometric\"", "extra_scores": {}, "extra_text": {}}]

Author rebuttal:

Meta-review: This paper present novel formulations to address the problem of unbalanced Gromov. The Conic formulation is very interesting but stays theoretical until optimization algorithms are available. The Unbalanced Gromov is a nice extension of Gromov and comes with relatively efficient solvers. Some very limited numerical experiment show the proposed UGW used between 2D distributions (two moons) and graphs.

The paper had some mixed reviews with reviewers acknowledging the novelty of the approach (albeit an extension similar to unbalanced OT) and of the theoretical results. The detailed a very well written response to the reviewers comment has been appreciated. But all reviewers also noted a lack of numerical experiments outside of the very simple illustrations in the paper. This paper is a very nice contribution to the theory of optimal transport but fails at illustrating its relevance to the ML community.  Despite acknowledging the theoretical contributions of the paper, the  AC recommends a reject but strongly encourages the authors to complete the experimental section with some ML applications or at least proof of concepts (graph classification, domain adaptation, ...). 

Decision: Reject