Title: Globally solving the Gromov-Wasserstein problem for point clouds in low dimensional Euclidean spaces
Venue: NeurIPS 2023
Accepted: yes
Keywords: Gromov-Wasserstein problem

QAP

Global optimization
Abstract: This paper presents a framework for computing the Gromov-Wasserstein problem between two sets of points in low dimensional spaces, where the discrepancy is the squared Euclidean norm.
The Gromov-Wasserstein problem is a generalization of the optimal transport problem that finds the assignment between two sets preserving pairwise distances as much as possible. This can be used to quantify the similarity between two formations or shapes, a common problem in AI and machine learning.
The problem can be formulated as a Quadratic Assignment Problem (QAP), which is in general computationally intractable even for small problems. Our framework addresses this challenge by reformulating the QAP as an optimization problem with a low-dimensional domain, leveraging the fact that the problem can be expressed as a concave quadratic optimization problem with low rank. The method scales well with the number of points, and it can be used to find the global solution for large-scale problems with thousands of points.
We compare the computational complexity of our approach with state-of-the-art methods on synthetic problems and apply it to a near-symmetrical problem which is of particular interest in computational biology.

Globally solving the Gromov-Wasserstein problem for point clouds in low dimensional Euclidean spaces

Abstract

This paper presents a framework for computing the Gromov-Wasserstein problem between two sets of points in low dimensional spaces, where the discrepancy is the squared Euclidean norm. The Gromov-Wasserstein problem is a generalization of the optimal transport problem that finds the assignment between two sets preserving pairwise distances as much as possible. This can be used to quantify the similarity between two formations or shapes, a common problem in AI and machine learning. The problem can be formulated as a Quadratic Assignment Problem (QAP), which is in general computationally intractable even for small problems. Our framework addresses this challenge by reformulating the QAP as an optimization problem with a low-dimensional domain, leveraging the fact that the problem can be expressed as a concave quadratic optimization problem with low rank. The method scales well with the number of points, and it can be used to find the global solution for large-scale problems with thousands of points. We compare the computational complexity of our approach with state-of-the-art methods on synthetic problems and apply it to a near-symmetrical problem which is of particular interest in computational biology.

1 Introduction

Many important applications in machine learning deal with comparing sequences, images, and higher dimensional data, where the data is unstructured and not directly comparable. In physics, chemistry, biology, music, and linguistics, objects with greatly different properties often appear in symmetrical variations characterized by concepts such as isomerisms, chirality, harmonies, and alternations. Understanding, and being able to analyze, these types of variations can be truly critical

as some variations in chemicals and biologicals may be toxic or even lethal. The Gromov-Wasserstein framework [15, 16, 17], has shown to be a powerful approach for comparing and matching such data, as it is invariant to translations and rotation. The Gromov-Wasserstein framework has, for example, been successfully applied to domain adaptation [29], graph matching [28], metric alignment [9], single-cell alignment [7], and word embedding [1].

The task of evaluating the Gromov-Wasserstein problem is in general considered to be intractable. Typically, the computational burden grows exponentially with the number of points describing the compared objects. In fact, a Gromov-Wasserstein problem can be formulated as quadratic assignment problem (QAP) [14, 5, 4], which is known to be NP-Hard. Naturally, there has been plenty of research on local and approximate methods for solving Gromov-Wasserstein and QAP problems [20, 23, 22, 27, 2, 24, 25]. However, objects containing symmetries or repeated patterns are particularly challenging for local optimization methods and may lead to significant errors in the estimated discrepancy as matching such objects with local optimization methods may accidentally find the sub-optimal reflections and rotations. The inability to detect such phenomena can have a great impact on the discovery of isomerisms and subsequently attributes of crucial importance.

In this paper, we develop a rigorous method for globally optimizing Gromov-Wasserstein problems by calculating a sequence of iteratively improving upper- and lower bounds. We consider a general class of Gromov-Wasserstein discrepancy problems where the points, representing the objects, belong to a Euclidean space. We show that such Gromov-Wasserstein problems can be formulated exactly as low-rank QAPs. We build upon this low-rank QAP representation to develop an algorithm that scales well with the number of points. The proposed algorithm can be characterized as a so-called cutting plane method [12, 10] where we solve a sequence of relaxed problems that are iteratively strengthened by generating and accumulating valid linear inequality constraints, i.e., cutting planes. The optimum of the relaxed problem provides a valid lower bound for the optimum of the Gromov-Wasserstein problem in each iteration. By solving a computationally cheap optimal transportation problem [18, 26, 6], we obtain both an upper-bound and a new cutting plane to strengthen the relaxation. We prove convergence for the proposed algorithm, and present a computational study that clearly shows the algorithm's efficiency and that the performance scales well with the number of points.

The main contribution of the paper can be summarized as:

We identify a general class of Gromov-Wasserstein problems, for point clouds embedded in low dimensional Euclidean spaces, that can be exactly represented as a concave low-rank QAP. In particular, mappings of images fits well within our framework.

We develop a method for solving this class of Gromov-Wasserstein problems by solving a sequence of alternating sub-problems, which are either low-dimensional or linear.

We prove that the proposed algorithm converges to a global optimal solution. The algorithm produces an optimality certificate in each iteration, in the form of upper- and lower bounds, which informs us of the potential suboptimality if the algorithm is terminated early.

Wepresent a numerical study, showing the efficiency of the proposed algorithm by comparing to other global optimization methods. We also illustrate the importance of globally solving Gromov-Wasserstein problems on a problem in computational biology.

In Section 2 we introduce the Gromov-Wasserstein problems and how it can be written as a QAP. In Section 3 we identify a class of Gromov-Wasserstein discrepancy problems that can be written as a concave relaxed QAPs problem, and in Section 4 we present the main methodology and an algorithm for solving this class of problems. Finally, in Section 5 we present numerical results and an application in computational biology.

2 The Gromov-Wasserstein discrepancy problem

Let x 1 . . . , x n ∈ X and y 1 . . . , y n ∈ Y be two sets of points and consider the problem of finding an assignment π between the point sets such that the pairwise distances d X ( x i , x i ′ ) and d Y ( y π ( i ) , y π ( i ′ ) ) are as close as possible for i, i ′ = 1 , . . . , n , where d X and d Y represents a notion of distance on the sets X and Y , respectively. This can be formulated as the discrete Gromov-Wasserstein discrepancy

problem

and where the assignment π is represented by a permutation matrix Γ and P is the set of all n × n permutation matrices. The corresponding relaxed problem, where instead Γ is in the set of doubly stochastic matrices, denoted by P , is often referred to as the Gromov-Wasserstein problem [20]. In these formulations we note that

where C x = [ d X ( x i , x i ′ )] n i,i ′ =1 , C y = [ d Y ( y j , y j ′ )] n j,j ′ =1 , and ⟨· , ·⟩ denotes the standard (Frobenius) inner product. Since the first and third sums are independent of Γ , solving the discrete GromovWasserstein problem (1) is the same as solving a quadratic assignment problem (QAP) on a simplified Koopmans-Beckmann form [5], namely as

This problem is in general NP-hard, and the number of variables scales with the number of data points, making (2) computationally intractable for problems of relevant size. Here, we focus on instances where the matrices C x , C y are positive definite and low rank. By utilizing this structure, we develop an algorithm that is guaranteed to find a globally optimal solution and scales well with the number of points.

3 The Gromov-Wasserstein problem and low rank QAP

An important special case of the Gromow-Wasserstein problem, considered in [20, 22], is when the point clouds belong to the Euclidean space and the squared Euclidean distance is used as discrepancy. That is, when the set of points are x 1 . . . , x n ∈ R ℓ x and y 1 . . . , y n ∈ R ℓ y , which we represent by the matrices

In this case it can be noted that the distance matrices C x and C y has a rank bounded by ℓ x +2 and ℓ y +2 , respectively, which can be seen from the identities

where m x = ( ∥ x 1 ∥ 2 , ∥ x 2 ∥ 2 , . . . , ∥ x n ∥ 2 ) T , m y = ( ∥ y 1 ∥ 2 , ∥ y 2 ∥ 2 , . . . , ∥ y n ∥ 2 ) T , and 1 ∈ R n × 1 is a column vector of ones. This observation was also used in [22] for formulating the GromovWasserstein problem as a quadratic problem of rank ( ℓ x +2)( ℓ y +2) and developing fast algorithms for the problem. However, the rank can be even further reduced and the corresponding GromovWasserstein problem can be formulated as a QAP problem of rank ℓ x ℓ y [25, Lemma 4.2.3] (cf. [20, Proposition 1]).

Proposition 1. [25, Lemma 4.2.3] Let Γ be a doubly stochastic matrix and the matrices C x and C y given by (3) , then it holds that

Proof. The proposition follows by straightforward computations, where the expressions are simplified using 1 = Γ 1 = Γ T 1 . See the appendix for the proof.

Hence, the low rank QAP formulation of the discrete Gromov-Wasserstein problem can be stated as

where L = 2 nm x m T y -4 m x 1 T Y T Y -4 X T X 1 m T y , and c 0 = ( ⟨ C x , C x ⟩ + ⟨ C y , C y ⟩ -4 1 T m y 1 T m x ) / 2 . The Gromov-Wasserstein problem can also be rewritten similarly and formulated as

and since the objective function is concave, any optimal solution of (4) is also an optimal solution of the relaxed problem.

Proposition 2. Any optimal solution of the discrete Gromov-Wasserstein problem (4) , is also an optimal solution to the Gromov-Wasserstein problem (5) . Conversely, problem (5) always has an optimal solution in one extreme point, 1 and any optimal extreme point to (5) is also an optimal solution to (4) .

Proof. Since (5) is the minimization of a concave objective function over a convex sets P , it attains the optimal value in an extreme point of the feasible set. Since the permutation matrices are the extreme points to the doubly stochastic matrices, i.e., P = ext( P ) , (5) attains its minimum on P . Further, the set of points in P for which (5) attains its minimum are the optimal solutions of (4). To show the converse statement, note that a minimum exists since P is compact and the objective function is continuous. Further, since the objective function is concave, an optimum must be at an extreme point. Finally, since the extreme points of P is the permutation matrices P , any optimal extreme point of (5) is also feasible and optimal to (4).

In the next section we will propose a methodology and an algorithm for solving this problem.

4 A cutting plane algorithm utilizing the low rank structure

By Proposition 2, we know that an optimal solution to the discrete Gromov-Wasserstein problem (4) can be obtained by solving the relaxed problem (5). However, the relaxed problem (5) is still a high-dimensional non-convex QP, which is NP-hard [19]. The high dimensionality can, in particular, be a limiting factor in solving the problem. For example, it is known that the performance of spatial branch-and-bound, one of the main approaches for globally optimizing nonconvex problems [10], can scale poorly with the number of variables. Thus, directly optimizing either (1) or (5) by spatial branch-and-bound is not computationally tractable for larger instances. Our idea is to use the low-rank formulation of the Gromov-Wasserstein problem and perform the optimization in a projected subspace of dimension ℓ x ℓ y +1 by solving a sequence of relaxed problems.

First, we note that problem (5) can be written as

Equivalence of problems (5) and (6) is shown by simply inserting the expressions for w and W into the objective function. Next, we project out the Γ variables, and we define the feasible set in the ( w,W ) -space as F = Proj W,w ( W ∈ R ℓ x × ℓ y , w ∈ R , Γ ∈ P ∣ ∣ W = 2 X Γ Y T , w = ⟨ L, Γ ⟩ ) .

Constructing an H-representation of the polytope F , i.e., representing it by linear constraints of the form ⟨ Z r , W ⟩ + α r w ≤ β r , is not trivial and the number of constraints can grow exponentially with the number of data points. Therefore, we propose an algorithm based on a cutting plane scheme to optimize over F .

Instead of directly optimizing the objective in (6a) over the feasible set F , which we don't have a tractable representation for, we relax the problem as

The linear constraints (7b) are supporting hyperplanes of the feasible set F , which we will generate iteratively. The goal is to force the minimizer of problem (7) into the feasible set F by using relatively few linear constraints. Keep in mind, we don't need a full representation of set F , we only need to capture the shape of F in some areas of interest, e.g., the constraints defining the faces of F at the optimal solution of problem (6) would suffice. The main advantage of the relaxation in problem (7) is that it contains far fewer variables than both problems (5) and (6), and the dimensionality is independent of the number of data points. Problem (7) can, therefore, be solved much more efficiently, especially in early iterations when the number of constraints is low. We will show that the constraints can be determined, as needed, by solving optimal transport problems. Based on this, we will develop an iterative approach that sequentially solves problem (7) and adds a constraint until the the solutions is the same as (6).

To initialize the search we determine a bounding box of F and use this to define a set of constraints (7b). The bounding box is determined by the (elementwise) minimum and maximum of the variables w and W given by

which can each be computed efficiently by solving a standard optimal transport problem. Initializing the set of constraints by the bounding box ensures that (7) is well-defined and bounded.

If the minimizer of problem (7) is within F , then we can stop as the solution is optimal for (6). 2 Otherwise, we improve the outer approximation of F by adding new a constraint defined by Z N +1 , α N +1 and β N +1 . Let ( w N , W N ) be the current optimal solution of (7), and assume that ( w N , W N ) / ∈ F , then we form a new constraint, a so-called cutting plane, that excludes ( w N , W N ) from the feasible set of (7).

We form a new constraint based on the gradient of the objective function (7a), which is given by

By letting α N +1 = 1 and Z N +1 = 2 W N , the hyperplane defining the new constraint will have the (negative) gradient in the optimum ( w N , W N ) as normal vector. Then we select β N +1 such that the new constraint forms a supporting hyperplane of F (7b). This can be found by solving the following optimal transport problem

When solving this problem, we also obtain a solution Γ N which is a doubly stochastic matrix (generically also a permutation matrix), which gives an upper bound for (6) and a candidate for the optimal solution. In the following subsection, we prove that the that algorithm, described in Algorithm 1, converges to a globally optimal solution.

A geometrical illustration of the algorithm is given in Figure (1). For illustrative purposes, we have used one-dimensional data resulting in a two-dimensional problem in the ( W,w ) -space. The data sets consist of 6 points each where one of the data sets has a reflective symmetry. This results in two global optima and 6! projected permutations in P . The first solution of (7) is located at one of the corners of the bounding box and marked with a "1" (the subsequent solutions are marked "2" "5"). The infeasible point "1" is excluded from the search space by a cutting plane (red line, marked with an "A"). Following the same procedure we obtain point "2", and cutting plane "B". Adding further cutting planes excludes "3" and subsequently "4", resulting in the feasible and optimal point "5". Note that the cutting planes from iterations 3 and 4 almost overlap the cutting planes "A" and "B", since the gradients in the points 1 and 3 are very similar (the same for points 2 and 4).

Algorithm 1 Gromov-Wasserstein problem

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4.1 Proof of convergence of Algorithm 1

The main result considering convergence is presented in the following theorem.

Theorem 1. The gap between the upper bound and lower bound in Algorithm 1 converges to 0 (if the tolerance is ϵ = 0 ).

Proof. Consider the N th iteration in Algorithm (1), let ( w N , W N ) be an optimal solution to (7), and Γ N is an optimal solution to (9) with corresponding points ( ˆ w, ˆ W ) = ( ⟨ L, Γ N ⟩ , 2 X ˆ Γ N Y T ) , in the ( w,W ) -space. Assume that the gap in the objective function between those two points is

The new constraint is then defined by ⟨ Z N +1 , W ⟩ + w ≤ β N +1 where Z N +1 = 2 W N and β N +1 = 2 ⟨ W N , ˆ W ⟩ + ˆ w , and thus for any point ( w,W ) that satisfy the constraint it must hold that

/a0

By substituting ˆ w from (10), we obtain

where we in the last step have used the Cauchy-Schwarz inequality.

For any iteration number N + k with k > 0 , we have that ( w N + k , W N + k ) is feasible for ⟨ Z N +1 , W ⟩ + w ≤ β N +1 , and thus the Euclidean distance between ( w N , W N ) and ( w N + k , W N + k ) is at least ϵ N / (1 + 4 ∥ W N ∥ 2 F ) . If the gap in the algorithm does not converge to 0 , then there is an ϵ > 0 for which ϵ N ≥ ϵ for all N and thus the distance between any two points in the sequence { ( w N , W N ) } N is bounded from below by ϵ/ (1+4max {∥ W ∥ 2 F | W ∈ (8) } ) . However, since the infinite sequence of points { ( w N , W N ) } N belong to a bounded set defined by (8), there must be a convergent subsequence, which contradicts that there is a positive lower bound on the distance between any two points.

From Theorem 1 the gap between the upper and lower bound converges to zero, and thus Algorithm 1 converges to a globally optimal solution.

4.2 Considerations when solving the relaxed problem

Problem (7) minimizes a concave function over a convex set. Thus, the solution is located in the extreme points of the convex set, i.e., the outer approximation of F . The standard approach to solve such problems is by branch and bound methods. However, the low dimension and sequential generation of constraints make it viable to search among the extreme points for an optimal solution.

To simplify notation, we define x T = ( w vec( W ) T ) ∈ R r where r := ℓ x ℓ y +1 . Then we can write (7b) on the form A k x ≤ b k . Note that, by construction, none of the constraints are strongly redundant as every constraint is satisfied with equality for a permutation. As the constraints are added sequentially, it is actually easy to compute the new extreme points by keeping track of previous extreme points as described in the following proposition.

Proposition 3. Assume that the extreme points { x k } k of the convex set described by Ax ≤ b are known. When adding a constraint A T N x ≤ b N , the additional extreme points are linear combinations of pairs of existing extreme points x k 1 and x k 2 both satisfying the same r -1 constraints with equality and A T N x k 1 ≤ b n and A T N x k 2 > b n so that the combination satisfies A T N ( λx k 1 +(1 -λ ) x k 2 ) = b N .

Proof. This is done by counting the number of constraints satisfied with equality. See the appendix for the proof.

Especially, in lower dimensions, e.g., with two or three dimensional data, this approach of keeping track of all extreme points and calculating new extreme points after adding a constraint can be very efficient for solving problem (7). More details of this method is provided in the appendix. In the numerical results, we present results where problem (7) is solved both by this extreme point search and the spatial branch and bound method in Gurobi.

5 Numerical results

5.1 Computational efficiency

In this section we compare the time to solve the problem up to an accuracy measured in relative error with different methods: Algorithm 1 when (7) is solved with the extreme point method as described in section 4.2, Algorithm 1 when (7) solved using Branch & bound using Gurobi 10.0 [13], MILP1 formulation in [11] implemented in Gurobi and finally when (6) is directly solved using Gurobi. The MILP1 formulation can handle a larger class of problems, but is reported to handle very few dimensions. All computations were performed using Matlab on an Intel i5 2.9 GHz PC. The linear optimal mass problem (9) was solved using the package [3] which is based on the network simplex [18]. The model problems tested are evenly distributed points in a unit disc or ball which we denote

U , and normally distributed points N (0 , σ ) . We denote N 1 := N (0 , I ) , N 2 := N (0 , diag(1 , 1 , 1 10 )) and N 3 := N (0 , diag(1 , 1 2 , 1 10 )) . See Table 1 for numerical results. Some notes on the results

On 2-dimensional data ( ℓ x = ℓ y = 2 ), the extreme point method is particularly efficient.

For problems that need many extreme points ( > 10 6 ), which depends on the data itself, the handling of extreme points becomes the driver of computational cost.

Problems mainly containing reflections (e.g. N 3 ) are easier to solve than those with room for rotations.

Directly solving (6) with Gurobi was not feasible for problems with n ≥ 500 .

5.2 Comparison with local search method

We compare the results using the proposed method to a local search method provided in the Git-hub repository for [20]. The entropy regularization parameter is set to 0 , and the method is run with random initializations (including the first lower bound [17] used with success in [22]) until the relative error to the global optimum is less than a specific tolerance ϵ . The problems are the same as in the previous section. Note here that in the local search methods we need an oracle in order to determine when we have reached a given performance level (which of course is not available in practice), whereas we in the proposed method computes upper and lower bounds.

Results for two and three dimensional data ( ℓ x = ℓ y = 2 , 3 ) are presented in Table 2. The results show that the proposed method performs better than multi-starting the local method when ℓ x = ℓ y = 2 . For the matching of 2-dimensional data to 3-dimensional data, the local search method is surprisingly fast suggesting that the problems is of a completely different nature than when 2-d data is matched to 2-d data or 3-d data is matched to 3-d data.

Note that we have chosen to compare with the method from [20] rather than [22]. This is since we have not optimized the optimal transport computations using the low rank structures in the problem. If we optimized the computations in this way, i.e., as in [21], we expect to get similar improvement as in [22] compared with [20].

5.3 Application to symmetrical data for morphological analysis

In this example we investigate the impact of correctly evaluating the Gromov-Wasserstein discrepancy compared to estimating it by local search. As a test case, we examine the ability to classify Adeno Associated Viral (AAV) particles based on the Gromov-Wasserstein discrepancy on image data originating from transmission electron microscopy, hence ℓ x = ℓ y = 2 . AAV particles are nearly round viral particles with multiple near-rotational symmetries as illustrated in Figure 2. By sampling n = 500 positions on each AAV particle proportional to the protein density, the point sets from pairs of particles X i and X j can subsequently be compared using the Gromov-Wasserstein discrepancy.

Computing the Gromov-Wasserstein discrepancy between all objects in a large set X = { X i } N i is tedious. Therefore, one may consider calculating the discrepancy to a subset of the objects that are well distributed under the Gromov-Wasserstein discrepancy. To find such a subset without actually calculating all pairs of discrepancies, we use a greedy approach by defining the index subset S k = { s i } k i =1 by selecting the first object arbitrarily and then let the set grow by

In this way all objects are closer than a tolerance to an object in the subset, and the way the subset is produced generates a monotonically decreasing tolerance. By using this procedure, every object obtains a feature vector of distances to the objects indexed by S k . Next the feature vectors are used as input to a K-means clustering and classification quality in terms of purity, adjusted rand index and normalized mutual information compared to an expert evaluation is presented in Figure 3.

The example shows that if the data contains symmetries, local search methods may get stuck on permutations that are locally optimal but that are far from globally optimal, see Figure 2. Thus the discrepancy obtained from local search is inappropriate to use as a discriminating feature, as shown in the lower performance in the classification illustrated in Figure 3. In the left subfigure of Figure 2, self-similarities are visited in the proposed method, and these local optima are in fact also almost optimal when used as initiation point using local search methods e.g., [20].

This example shows that when the Gromov-Wasserstein problem is calculated accurately it provides valuable information and biological meaning as it differentiates viral particles with different cargo and

variations in capsid structure, and, at the same time, finds the optimal orientation positively revealing possible chiralities or isomerisms. It also shows that when the distance is calculated accurately, it provides better decision support than using local search methods.

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6 Discussion

When using distances as input for statistical analyses, the accuracy of the measurement set a bound for the information resolution. If the measurement system introduces error of a certain structure, this can produce artefacts in the result and affect decisions taken on the result. When using distances for such purposes, it is necessary to either know the measurement error, the artefacts being produced, or using an accurate measurement system. In this paper we have provided a method which computes the Gromov-Wasserstein problem accurately, which reduces the uncertainty of such considerations.

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Hongteng Xu, Dixin Luo, and Lawrence Carin. Scalable Gromov-Wasserstein learning for graph partitioning and matching. Advances in neural information processing systems , 32:3052-3062, 2019.

Hongteng Xu, Dixin Luo, Hongyuan Zha, and Lawrence Carin Duke. Gromov-Wasserstein learning for graph matching and node embedding. In International conference on machine learning , pages 6932-6941. PMLR, 2019.

Yuguang Yan, Wen Li, Hanrui Wu, Huaqing Min, Mingkui Tan, and Qingyao Wu. Semisupervised optimal transport for heterogeneous domain adaptation. In IJCAI , volume 7, pages 2969-2975, 2018.

A Appendix

A.1 Proof of Proposition 1

Proposition 4. Let Γ be a doubly stochastic matrix and the matrices C x and C y given by

then it holds that

Proof. The proposition follows by the following straightforward computations, where we just expand the expressions and use 1 = Γ 1 = Γ T 1 , thus

A.2 Proof of Proposition 3

Proposition 6. Assume that the extreme points { x k } k of the convex set described by Ax ≤ b are known. When adding a constraint A T N x ≤ b N , the additional extreme points are linear combinations of pairs of existing extreme points x k 1 and x k 2 both satisfying the same r -1 constraints with equality and A T N x k 1 ≤ b n and A T N x k 2 ≥ b n so that the combination satisfies A T N ( λx k 1 +(1 -λ ) x k 2 ) = b N .

Proof. Let W be a matrix whose columns consist of the extreme points defined by the N -1 constraints Ax ≤ b . Also, let a T N x ≤ b N be an additional constraint, e k be a unit vector with 1 on position k , and let α parametrize the convex cone on W , i.e. 1 T α = 1 , α ≥ 0 so that AWα ≤ b describes all points in the convex set. Suppose that B k describes the indices of the constraints that define the k:th extreme point by letting A B k be the sub matrix of A including the rows denoted by the indices in B k . Then A B k We k = b B k . It then follows that AW ( λe k 1 +(1 -λ ) e k 2 ) j = b j ifand only if j ∈ B k 1 and j ∈ B k 2 and 0 ≤ λ ≤ 1 . The construction in the proposition, λx k 1 +(1 -λ ) x k 2 so that A T N ( λx k 1 +(1 -λ ) x k 2 ) = b N then satisfies r constraints with equality and all other constraints with inequality, i.e. the point is an extreme point to the set. We also note that every multiple combination (three or more) of extreme points sharing r -1 constraints are not extreme points as they are linear combination of the pairs given by α ( λ 1 x 1 +(1 -λ 1 ) x 2 ) + (1 -α )( λ 2 x 2 +(1 -λ 2 ) x 3 ) . Every linear combination of pairs of points sharing less than r -1 constraints will not satisfy r constraints, i.e. they are not extreme points.

A.3 A description of the implementation of the extreme point method

The handling of the extreme points in the paper is done by keeping track of the extreme points, their connection to the boundary constraints, and lookup tables for the adjacent extreme points, i.e., extreme points that satisfies the same r -1 constraints with equality, where r is the rank of the problem. Let the extreme points be described in the matrix E where each column describes an extreme point. Thus AE ≤ b 1 , if the constraints are described by the matrix A and vector b such that Ax ≤ b is the constraint equations.

Let the adjacency be described in a (sparse) matrix with binary elements D where D i,j = 1 if extreme point i and j are adjacent. Let us also keep track of the constraints that are satisfied by an extreme point with equality. For this purpose let B be a matrix with binary elements in which the element B i,j = 1 if extreme point j satisfies constraint i with equality. Thus, D i,j = 1 if ( B T B ) i,j = r -1 . This is one of the computational drivers for the proposed extreme point method.

When a new constraint ( A n , b n ) is added, the new extreme points are generated by a linear combination of the infeasible extreme points I := { i : A n E i > b n } and their adjacent feasible extreme points f i := { j : D ( i, j ) = 1 } where i ∈ I . Let # indicate the number of elements of a finite set, then ∑ i ∈I #( f i ) is the number of new extreme points. We place the new extreme points in the matrix E by adding them in the end as a matrix P . The new matrix containing the extreme points

The matrix keeping track of which extreme points satisfies which constraints with equality is extended with

where C · ,k = B · ,i ⊙ B · , ( f i ) k , preferably implemented using bitwise operators. Here, the last row describes the newly added constraint and k a re-enumeration of the new extreme points.

The new adjacency matrix D n can be concatenated with the old D and two additional matrices

where O i,j = 1 the old extreme point i is adjacent to the new extreme point j . This information is already available for us, since the new extreme points are adjacent to f j . Finally, N i,j = 1 if ( C T C ) i,j = r -2 . This is by far the most computationally expensive operation in the proposed algorithm, which can be implemented with std::popcount in the standard c++ library. For 3-dimensional problems, around 120 bits needs to be compared between all new extreme points.

A.4 Additional tests

A.5 Convergence rate

The convergence proof in Proposition 3 does not include a rate of convergence. In Figure 4, we show the convergence trajectory for the problems U and N 1 for 2-dimensional data. The tests show that the convergence rate is linear to its nature up to a number of iterations where the gap closes completely.

In comparison to the result of the proposed method we present the same convergence evaluation with the local search method in [20]. Figure5 shows the trajectory of convergence. The tests show that the rate of convergence is sublinear to its nature. The number of initializations needed to achieve a pre-determined accuracy increase with the number of points for the local search. Note here that in order to determine when to stop one needs to know the optimal value, which is not available for the local search method.

0

50

100

150

Iterations

10 -4

10 -2

10 0

Relative error

n=100

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100

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Iterations

10 -5

10 0

n=200

0

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400

Iterations

10 -5

10 0

n=300

0

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Iterations

10 -5

10 0

n=400

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50

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150

Iterations

10 -5

10 0

Relative error

n=100

0

100

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Iterations

10 -5

10 0

n=200

0

100

200

Iterations

10 -5

10 0

n=300

0

100

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10 -5

10 0

n=400

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100

initializations

10 -5

10 0

Relative error

n=100

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500

1000

initializations

10 -4

10 -2

10 0

n=200

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initializations

10 -6

10 -4

10 -2

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initializations

10 -6

10 -4

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n=400

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initializations

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A.6 Evaluation on the MNIST dataset

In this section we present the performance of the proposed method on shapes originating from the MNIST dataset [8]. Even though the arabic numerals are not entirely unique to reflections (e.g. "2" and "5") and rotations (e.g. "6" and "9"), the ability to compute the exact Gromov-Wasserstein discepancy is of interest. When the images doesn't contain the same number of points, evenly distributed points are taken from the image with more points. The maximum number of points was set to 400. Number of iterations are presented in Table 3 and the computational time is presented in Table 4, showing that the proposed method works as anticipated on the data set. Examples of correspondances of numerals are shown in Figure 6.

0

1

2

3

4

5

6

7

8

9

0

92.5

55.5

98.5

75.5

115.5

106.5

80

81

90

90

1

21

46.5

36

70.5

50.5

47

41

50

49

2

52

53.5

64

58.5

60.5

57.5

58

43

3

40.5

75

52.5

72.5

46.5

56

56.5

4

84.5

57

63.5

65

73.5

61.5

5

50

53.5

48

64.5

49.5

6

44

45.5

67

56.5

7

48.5

64

47.5

8

62.5

55.5

9

52.5

0

1

2

3

4

5

6

7

8

9

0

67-134

39-80

67-223

51-139

86-184

69-131

62-97

63-143

73-102

53-163

1

10-39

32-78

30-64

49-98

36-61

35-88

27-69

43-59

27-93

2

29-77

36-78

37-75

44-88

47-95

38-72

43-86

36-83

3

29-67

31-116

47-60

42-89

27-77

48-85

27-87

4

50-107

40-111

40-111

52-92

47-108

50-80

5

31-74

45-88

43-87

54-94

36-97

6

26-80

36-88

54-123

32-94

7

24-76

52-75

30-87

8

40-82

42-71

9

29-65

0

1

2

3

4

5

6

7

8

9

0

0.85

0.48

1.03

0.76

1.09

1.33

0.83

0.81

0.98

0.95

1

0.05

0.18

0.15

0.32

0.22

0.18

0.15

0.3

0.29

2

0.34

0.33

0.41

0.39

0.46

0.37

0.47

0.34

3

0.31

0.7

0.41

0.64

0.39

0.52

0.51

4

0.85

0.61

0.6

0.67

0.86

0.7

5

0.53

0.62

0.49

0.75

0.5

6

0.54

0.49

0.96

0.66

7

0.61

0.84

0.59

8

0.77

0.74

9

0.92

0

1

2

3

4

5

6

7

8

9

0

0.5-1.4

0.3-1

0.6-2.4

0.5-1.5

0.9-1.9

0.6-1.6

0.6-1.2

0.5-1.7

0.6-1.4

0.5-2.1

1

0-0.1

0.1-0.4

0.1-0.4

0.2-0.4

0.2-0.3

0.1-0.4

0.1-0.4

0.2-0.4

0.1-0.5

2

0.2-0.5

0.2-0.6

0.2-0.5

0.3-0.6

0.3-0.8

0.3-0.5

0.3-0.7

0.2-0.7

3

0.2-0.7

0.2-1.2

0.3-0.5

0.3-0.9

0.2-0.7

0.4-0.7

0.2-0.8

4

0.4-1

0.3-1

0.3-1.3

0.4-0.8

0.4-1.3

0.4-0.9

5

0.2-0.9

0.4-1

0.3-0.9

0.5-1.1

0.4-1.1

6

0.2-1

0.3-1

0.6-1.8

0.3-1.2

7

0.3-0.9

0.7-1.2

0.1-1.4

8

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0.5-1.1

9

0.5-1.1

Peer reviews:
{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThis paper consider the Gromov-Wasserstien (GW) problem with quadratic cost, a (non-convex) quadratic optimization problem over the space of probability measures (in this work, restricted to uniform discrete measures supported on $n$ points) which takes the form (in discrete setting):\n$$\\min_{\\Gamma} \\sum_{1 \\leq i,j \\leq n} ||x_i - x_{i'}|^2 - |y_j - y_{j'}|^2|^2 \\Gamma_{ij} \\Gamma_{i'j'},$$\nwhere $\\Gamma$ is a doubly stochastic matrix of size $n \\times n$. \n\nThis problem is known to be hard to solve globally. This paper proposes an approach to do so that is tractable when the point clouds $X= (x_1,\\dots,x_n)$ and $Y = (y_1,\\dots,y_n)$ are in low dimension---say $d$ [edit: fix rendering]. The core idea is that the GW problem can be reparametrized as \n\n\\begin{equation}\\tag{1}\n\\min_{(W,w), \\Gamma} - |W|^2 - w + c_0,\n\\end{equation}\nwhere $\\Gamma$ is still a bistochastic matrix, $W,w$ must satisfy the relation $W = 2 X \\Gamma Y^T$ and $w = \\braket{L,\\Gamma}$, where $L$ and $c_0$ are constant term (wrt $w,W,\\Gamma$). Note that $W$ can be understood as the correlation between $X$ and $Y$ wrt the joint law $\\Gamma$, which is of size $d \\times d$, so losely speaking, the quadratic part of the GW problem does not depend on $\\Gamma$ (which is of size $n \\times n$) but only on the correlation it induces (which is a much smaller object when $d$ is small). \n\nFrom this reformulation, the idea of the paper to globally solve GW is the following: \n- Let $\\mathcal{F}$ denote the constraint polytope which links $W,w$ and $\\Gamma$. This space is too complex to be represented explicitly (by linear constraints). \n- Build a bounding box of $\\mathcal{F}$. This is doable because we know that the correlation matrix between $X$ and $Y$ has (explicit) bounds on its entries---for instance when $d=1$ this is simply the cost of the increasing (resp. decreasing) matching. \n- Build iteratively a sequence of \"lower approximation\" $(H_N)_N$ of $\\mathcal{F}$ (made of supporting hyperplanes of $\\mathcal{F}$), in sense that minimizing the functional in (1) over $H_N$ yields a lower bound $L_N$ for the GW problem. Note: this is a concave minimization problem, which is tractable in low dimension ($d \\times d + 1$ here). \n- Let $(W_N, w_N)$ be the solution on $H_N$. Typically $W_N,w_N \\not\\in \\mathcal{F}$ (otherwise, we have found a global minimizer of GW), but from this we can build a new constraint $H_{N+1}$ through solving a standard OT problem, providing a doubly stochastic matrix $\\Gamma_N$ which is by definition sub-optimal, hence giving an upper-bound $U_N$ for GW. \n- Eventually, the authors prove that $U_N - L_N \\to 0$ as $N \\to \\infty$, so the proposed approach yields a practical (in low-dimension) algorithm that converges to a **global** solution of the GW problem.\n\nSTRENGTHS:\nThe paper considers the difficult (and important, in my opinion) question of globally solving the GW problem. \nThe proposed approach is based on existing ideas (parametrization of GW by low-dimensional matrices) but pushes them further to get an original and interesting approach. Having globally converging algorithms for GW, even if restricted to low dimensional setups with \"not-so-many points\", can be useful to assess the quality of other algorithms that may only converge locally (maybe some of them still converge globally \"most of the time\", etc.). \n\nThe paper is clear and does not sacrifice mathematical technicality. Proofs of Prop 2 and Thm 1 have been checked and no major flaw was identified (aside from small details). \n\nI also appreciate that the paper immediately acknowledges that its approach is limited to low-dimensional problems (which is not a major issue; using GW in low dimension is completely natural).\n\nWEAKNESSES:\n# 1. On the convergence of the algorithm\n\nFrom my understanding, the proof of global convergence of the algorithm (Theorem 1) is purely asymptotic: the gap $\\epsilon_N = |U_N - L_N|$ is controled by $|\\theta_N - \\theta_{N+k}|,\\ (\\forall k)$ where $\\theta_N = (W_N, w_N)$ is compactly supported, which implies by contradiction that $0$ must be the single accumulation point of $\\epsilon_N$. \n\nWhile this makes sense (up to few technical considerations, see below), this can be considered as a weakness (at least from a theoretical viewpoint): the convergence of $\\epsilon_N \\to 0$ is controlled by \"how fast $\\theta_N$ accumulates\", and even in low dimension, without further investigation (i.e. thinking that $\\theta_N$ moves arbitrarily in a compact set), it may take a lot of time to reach low gap $\\epsilon$ (and this seems to be suggested by the experiments, where the criterion is set to $10^{-8}$ when $d=2$, but to $10^{-2}$ when $d=3$). \n\nI am not saying that this is what happen in practice, nor that this invalidates the contribution of this theoretical result, but a discussion on this may be welcome. \n\n\n# 2. On numerical experiments\n\nWhile the proposed experiments are conducted in a reasonable way, they remain somewhat limited in my opinion. In particular, \n- [related to the point above] I would have appreciated to have more illustrative experiments on the algorithm behavior, its convergence, etc. \n- As far as I can tell, the paper only compares with the work \"local search\" [16] (Peyré et al., 2016). Why not comparing with more modern works, as [17] (Scetbon et al., 2022), [D] by Sejourné et al. (2021) or [E] (Li et al., 2023)? Note that [17] seems to handle larger instances ($n=10^5$ points), but may fail to globally converge as far as I can tell. Showcasing the strengths of the current work vs [17] (probably the closest work in its spirit), even in illustrative scenarios, would be of interest. \n\n\n\n# Minor comments (rather suggestions than actual weaknesses)\n\n1. In Algorithm 1, the upper bound is updated as $U_{N+1} \\leftarrow \\min(U_N, \\text{OT cost} )$. From my understanding, this makes $U_N$ non-increasing, while $L_N$ is increasing (as a minimization problem with more and more constraints), so that $\\epsilon_N$ is decreasing. This is never mentioned as far as I can tell; but it seems to be used in the proof when saying that \"if $\\epsilon_N \\not\\to 0$, then it must be lower bounded\". Am I correct? Also, the proof says \"assume that $\\epsilon_N = \\dots$\" (which to me means \"assume that $U_{N+1} = \\text{OT cost}$ rather than $U_{N+1} = U_N$\"), but never discusses the possibility that $U_{N+1} = U_N$. In any case, this does not invalidate the proof as, from my understanding, the $\\min$ in the algorithm is not required to prove convergence (even if we cannot ensure that $\\epsilon_N$ is decreasing). But in any ways, I think that this is worth some clarification. \n\n2. The discussion related work and context can be improved. For instance, I do not think that [16] is a suited reference in the introduction with mentioning the GW problem for the first time. It would rather be credited to either Mémoli (2011), or Sturm (2012 - \"The space of spaces\"). What is new vs known (from [16], or for instance from [A, Sec 2.2.3] and related works) in sections 2 and 3 should also be highlighted. Similarly, using that the quadratic term in GW only depends on $X \\Gamma Y^T$ is also used (among other) in [B, C] (note : [C] is a preprint put on arxiv after neurIPS submission, this is a suggestion for the revised version, not a criticism). \n\n3. [typo] line 90 : \"has\" should be \"have\" I think?\n\n4. [typo] line 114, I think that \"maximum\" should be \"minimum\" ? \n\n5. I wonder how useful are the variables $Z_N, \\alpha_N$, given that $Z_N$ is simply $2 W_N$ and $\\alpha_N = 1$. I understand that this is a convenient way to write a \"general\" hyperplane equation $\\braket{Z,W} + \\alpha w \\leq \\beta$, but to me it turned out to hinder the reading a bit. \n\n6. [typo] line 168, \"that the that\". \n\n\n# References\n\n- [A] A contribution to Optimal Transport on incomparable spaces, T. Vayer, 2020\n- [B] On the existence of Monge maps for the Gromov-Wasserstien problem, Dumont et al., 2023.\n- [C] The Gromov-Wasserstein distance between spheres, Arya et al., 2023. \n- [D] The Unbalanced Gromov Wasserstein Distance: Conic Formulation and Relaxation, Sejourné et al., 2021\n- [E] A Convergent Single-Loop Algorithm for Relaxation of Gromov-Wasserstein in Graph Data, Li et al., 2023\n\nQUESTIONS:\nI think that the work may be improved by adding: \n- A discussion / numerical illustration on the convergence (rate) of the algorithm (Section 1. in Weaknesses), or an explanation of why this is not relevant. \n- A more precise comparison with concurrent works, in particular [17] (which is the closest to the current work as far as I can tell), from both a theoretical and numerical perspective ; where scalability but also quality of the result (which, I guess, may favor the proposed approach). If the comparison is not meaningful, please explain why.\n\nLIMITATIONS:\nThe authors clearly state that the work is dedicated to low-dimensional point cloud, which is its main limitation. \n\nAside from that, I do not see any potential societal impact specific to this work.", "TITLE": "Official Review", "RECOMMENDATION": 6, "REVIEWER_CONFIDENCE": 4, "SOUNDNESS_CORRECTNESS": 3, "DATE": "2023-06-26"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThe Gromov-Wasserstein optimal transport problem is a non-convex problem known to be hard, closely related to QAP. The authors consder its special case when the two involved metric spaces are Euclidean with small numbers of dimensions. We want to permute one set of points such that the sum of squares of differences of Euclidean distances between all point pairs is minimized. This simplification of this problem (compared to the general GW OT problem) is possible because the matrix of Euclidean pairwise distances for any set of points in $R^k$ has rank at most $k+2$. This allows the authors to formulate the problem as minimization of a simple concave quadratic function of a small ($kl+1$ where $k,l$ are the dimensions of the spaces) number of parameters over a convex polyhedral feasible set, which is a projection of the set of doubly-stochastic matrices. This problem is solved to global optimality by alternating two steps: (1) globally solve the concave minimization problem over an outer approximation of the feasible set, (2) improve this outer approximation by generating a cutting plane (a linear inequality valid for the above projection of the set of doubly-stochastic matrices). Step 1 can be done either by an off-the-shelf concave minimization algorithm (such as branch&bound) or by a reasoning over the set of extremal points of the feasible set (which is fast for low-dimensional spaces). Step 2 leads to solving the ordinary linear (Kantorovich) OT problem. The algorithm is proved to converge to a global optimum.\n\nThe proposed algorithm is first tested on synthetic data. Here, we match two randomly generated (either uniformly on a disc or normally distributed around the origin) point sets from $R^2$ or $R^3$. This is compared to other globally optimal methods solved by Gurobi and to the local search method [17] with multiple initializations. Second, the method is tested on a real application from biology. These experiments show that the method is almost always much faster than the other methods (the global methods are in fact usable only for small instances, due to the large number of parameters).\n\nSTRENGTHS:\nA simple algorithm for a difficult problem, without many tuning parameters, which performs well in experiments.\nClearly written.\n\nWEAKNESSES:\nThe most important issue is limited experimental testing. The main experiment is done on synthetic, randomly generated data. However, it is known that optimization algorithms often behave very differently on random data and on real data.\n\nMoreover, I do not quite understand the setup in the biological experiment, especially I get lost in the 1st paragraph in section 5.1. Please explain better. What is $n, l_x, l_y$ in this case?\n\nTo show more clearly the strengths and weaknesses of the approach, it should be tested on as many instance types as possible. Currently I am not convinced that the method would not last very (unacceptably) long for some instances from practice.\nOne option is to make more extensive synthetic experiments, with more complex data than just uniformly/normally distributed. E.g., one can take a non-random set of points in 2D or 3D (draw a shape or use a shape from a public shape database) and to synthetically generate the second set by permuting these points and adding noise to them (gaussian, uniform) or replace some of them with outliers.\n\nAnother obvious suggestion is shape/object matching on real data, where the point distances are measured by Euclidean metric (rather than geodesic). Possible inspirations for such experiments is [13, 17] (but, I believe, also other works).\n\nAs for novelty: Although the method just combines tools well-known in optimization, the main idea (decoupling the concave minimization part and the cutting plane part) is clever. Unfortunately, I do not know the relevant literature well enough to be sure that a similar approach has not been proposed before. I am surprised that no globally optimal algorithm for globally solving this problem has been proposed before by others - such algorithms have been apparently proposed only for general GW problem and QAP.\n\nThe initial formulation (1) of the GW problem seems to be wrong because, to my knowledge, the GW optimal-transport problem optimizes over doubly-stochastic matrices rather than permutation matrices (i.e., it allows \"soft matching\"). Therefore, the relaxation from (4) to (5) is unnecessary and in fact misleading. If (5) minimizes a concave function over a convex set, the two problems are indeed equivalent, but the explanation should go the other way.\n\nMinor issues / suggestions:\n\n- 77: \"the first two sums\" should be \"the 1st and 3rd sum\"\n\n- In (2), it would be logical to omit the 2nd and 3rd terms because they do not depend on Gamma (as noted above).\n\n- 92: You may wish to justify/cite why a Euclid distance matrix has rank at most $l_x+2$.\n\n- Proposition 1 should not be really a proposition, it is just a algebraic manipulation.\n\n- 133: The part defining the set $\\mathcal{F}$ is over-complicated and confusing. Why don't you write just \n$\\mathcal{F} = \\\\{ (2X\\Gamma Y, \\langle L,\\Gamma\\rangle) \\mid \\Gamma \\in P \\\\}$, which shows that $\\mathcal{F}$ is a linear map of $P$.\n\n- In the final version, Table 1 should report also the number $N$ of iterations (= added cutting planes) needed to achieve the prescribed accuracy. Currently, this is only in the supplement.\n\nQUESTIONS:\nWhy were experiments on more real instance types not included, such as with shape/object matching? Is there a substantial obstacle or you just considered this as out of the scope of this paper?\n\nCan the method be extended to be resilient to outliers? One option for this would be to use 1-norm rather than 2-norm in pairwise distances - but this would violate the low-rank assumption and make the approach inapplicable.\n\nCan the trick be applied to a wider class of problems, such as a wider subclass of QAP?\n\nLIMITATIONS:\nThe experiments are limited, not convincingly showing efficiency of the method on a wide enough class of real instances.", "TITLE": "Official Review", "RECOMMENDATION": 6, "REVIEWER_CONFIDENCE": 4, "SOUNDNESS_CORRECTNESS": 3, "DATE": "2023-06-29"}

comments: SUMMARY:
- The paper introduces a novel algorithm for efficiently computing the Gromov-Wasserstein (GW) distance for low-dimensional point clouds. The proposed method offers a transformative approach by transforming the GW distance problem into a sequence of concave optimization problems over convex sets, thereby improving computational efficiency.

- By leveraging this new algorithm, the authors conducted comparative evaluations against existing techniques for computing the GW distance. The results demonstrate significant improvements in terms of computational speed.

STRENGTHS:
- One of the key strengths of this paper is the introduction of a novel and interesting algorithm. The proposed method offers a fresh perspective on computing the Gromov-Wasserstein (GW) distance for low-dimensional point clouds. 
- The research holds significant importance in the field of machine learning and data science, given the growing popularity of the GW distance. By proposing a method that accelerates the computation of the GW distance, the paper addresses a practical need and offers a valuable contribution to the field.
- The clarity of presentation is a notable strength of this paper. The authors effectively communicate their ideas, methodologies, and findings, making the paper accessible to readers. 
- The experiments well support the acceleration effect of the proposed algorithms.

WEAKNESSES:
A potential weakness of the paper is the lack of a complexity analysis for Algorithm 1, which could be addressed to further enhance the paper.

QUESTIONS:
- line 190: in the last inequality, am I right that a square root is missing?
- line 196: Is there a typo in the lower bound?

LIMITATIONS:
N/A
TITLE: Official Review
SOUNDNESS_CORRECTNESS: 4
DATE: 2023-07-03

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThis paper solves the Gromov Wasserstein (GW) distance problem for squared Euclidean norm by considering a low-dimensional space on which the computation is performed, using a cutting-plane method. To this end, they write the GW problem as a low-rank optimization problem. Their algorithm is supported by theoretical convergence guarantees.\n\nSTRENGTHS:\nThis article makes it possible to compute the Gromov-Wasserstein distance efficiently. In a way, this paper extends the results of [17] which reformulated the Gromov-Wasserstein problem as a low-rank quadratic problem. The idea of using the cutting plane algorithm to efficiently solve this optimization problem in a projected low-dimensional subspace is new, and the proposed algorithm converges to a global optimal solution.\n\nWEAKNESSES:\nSolving problem (9) at each iteration of the algorithms remains computationally heavy, as it corresponds to solving an optimal transport problem with $\\Gamma$ a $n\\times n$ matrix.\n\nFor a more relevant contribution, and as the use of the low-rank quadratic structure of the Gromov-Wasserstein optimization problem has already been exploited in [17], the authors should compare their results with [17] in experiments, and not only with the local search method of [16].\n\nQUESTIONS:\nThe matrices $C_x$ and $C_y$ are assumed to be positive definite and of low rank. How can the low-rank assumption be ensured? What does this mean in terms of the point clouds $X$ and $Y$?\n\nCan the cutting plane algorithm trick be extended to the case where both point clouds have weights (i.e. a non-uniform discrete measure), and for which the probability measures do not have the same number of points (therefore the mass in the transport plan $\\Gamma$ could split)?\n\nLIMITATIONS:\nA comparison with more recent methods than [16] for computing the Gromov Wasserstein distance should be included.", "TITLE": "Official Review", "RECOMMENDATION": 6, "REVIEWER_CONFIDENCE": 3, "SOUNDNESS_CORRECTNESS": 3, "DATE": "2023-07-06"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThis paper proposes a new algorithm to solve the Gromov-Wasserstein problem between two sets of points in Euclidean spaces when the ground cost is the squared Euclidean norm. This is done by first reformulating the Gromov-Wasserstein problem as a low-rank QAP problem, \nthen relaxing the set of admissible couplings in a way that the optimal solutions of the obtained relaxed problem stays the same, and then using a cutting-plane method to solve the relaxed problem. The obtained algorithm provides at each iteration a lower bound and an upper bound on the value of the optimal solution, and it can be proven that the gap between those bounds theoretically converges to zero at infinity, implying that the proposed algorithm theoretically converges to a global optimum. The experiments offer a comparison in term of computational efficiency between the proposed algorithm and a more \"traditional\" algorithm, and emphasize the importance of converging to a global optimum when using Gromov for applied problems.\n\nSTRENGTHS:\nI enjoyed reading this paper. To the best of my knowledge, the proposed algorithm is new. \nBeside the proposed algorithm in itself, the paper highlights that most of algorithms which compute Gromov only converge to a local minimum, which is a problem that is often ignored in the literature. Moreover, when solving the\nGromov-Wasserstein problem with \"traditional\" algorithms, it is not even possible to know in practice whether the solution\nwe've converged to is a local or global optimum. Thus, proposing an algorithm that is guaranteed to converge towards the global optimum is a real strength in my opinion.\n\nWEAKNESSES:\n- As someone familiar with the Gromov-Wasserstein problem but not expert in optimization, I find this paper a bit difficult \nto read because some important steps in the mathematical reasoning are lacking of details and so, in my opinion, too much of the reasoning is left to reader.\n- I think the experiment section could be improved: the experiment of Figure 2 could be clearer, and it could be nice \nto have a synthetic experiment that exhibit a case where the proposed algorithm converges to the global optimum whereas the local search method remains stuck in a local optimum.\n\nQUESTIONS:\nThe following questions are related to the steps of reasoning that I find could be clearer:\n- 1/ What is the definition of the rank of a QAP problem? \n- 2/ Why do we need to relax Problem (4) to doubly stochastic matrices instead of permutation matrices before applying \nthe cutting-plane method? \n- 3/ Can you explain how to obtain the initial $ (Z_r,\\alpha_r,\\beta_r)_r $ from solving the bounding box problem? \n- 4/ I'm a bit confused about Equation (8a): are we looking for a different $ \\Gamma $ for each $ i,j$? The way it is written suggests we're looking for the same for all $ i $ and $ j $. \n- 5/ $ N $ stands for the current iteration number,  but also for the number of initial constraints. I find that confusing. \n\nExperiments:\n- 1/ In Figure 2 left: are the differents structure plotted are local optima obtained using a local search method? I'm not sure \nto understand Figure 2 right either: the blue is the result of your algorithm? can you explain why it is better than the result with the local search method and where do we see the comparison with the expert evaluation? \n- 2/ How the methods compare itself (in terms of computational efficiency) to approached methods (as for instance Entropic Gromov, sliced Gromov,etc) with several random initializations?\n\nLIMITATIONS:\nThe authors has not clearly discussed the limations of the algorithm. \nIn my opinion, the principal limitation is that the method remains computationally costly, and so, as this is the case for almost all methods that compute Gromov, it is still unusable in higher dimension or when the number of points is large.", "TITLE": "Official Review", "RECOMMENDATION": 6, "REVIEWER_CONFIDENCE": 4, "SOUNDNESS_CORRECTNESS": 3, "DATE": "2023-07-06"}