Tangential Fixpoint Iterations for Gromov-Wasserstein Barycenters

TL;DR

A known procedure for the determination of Fr\'echet means in Riemannian manifolds via tangential approximations in the context of GW is revisited and a relaxation of this fixpoint iteration is proposed and it is shown that it monotonously decreases the barycenter loss.

Abstract

The Gromov-Wasserstein (GW) transport problem is a relaxation of classic optimal transport, which seeks a transport between two measures while preserving their internal geometry. Due to meeting this theoretical underpinning, it is a valuable tool for the analysis of objects that do not possess a natural embedding or should be studied independently of it. Prime applications can thus be found in e.g. shape matching, classification and interpolation tasks. To tackle the latter, one theoretically justified approach is the employment of multi-marginal GW transport and GW barycenters, which are Fréchet means with respect to the GW distance. However, because the computation of GW itself already poses a quadratic and non-convex optimization problem, the determination of GW barycenters is a hard task and algorithms for their computation are scarce. In this paper, we revisit a known procedure for the determination of Fréchet means in Riemannian manifolds via tangential approximations in the context of GW. We provide a characterization of barycenters in the GW tangent space, which ultimately gives rise to a fixpoint iteration for approximating GW barycenters using multi-marginal plans. We propose a relaxation of this fixpoint iteration and show that it monotonously decreases the barycenter loss. In certain cases our proposed method naturally provides us with barycentric embeddings. The resulting algorithm is capable of producing qualitative shape interpolations between multiple 3d shapes with support sizes of over thousands of points in reasonable time. In addition, we verify our method on shape classification and multi-graph matching tasks.

Figures (4)

• Figure 1: Gromov--Wasserstein interpolations $(\mathbb{Y}_\rho)_{\rho \in \Delta_{1}}$ between two 3d shapes for a total of four input pairs $(\mathbb{X}_1,\mathbb{X}_2)$. From left to right, each row shows $\mathbb{X}_1, \mathbb{Y}_{(5/6,1/6)}, \mathbb{Y}_{(4/6,2/6)}, \mathbb{Y}_{(3/6,3/6)}, \mathbb{Y}_{(2/6,4/6)}, \mathbb{Y}_{(1/6,5/6)}, \mathbb{X}_2$. The colouring indicates the optimal GW plan between the inputs and interpolants. The Euclidean diameters (before applying the PCA) and PCA residuals are shown below each input/barycenter. The GW distances are (top to bottom): 0.07, 0.03, 0.06, 0.15.
• Figure 2: A GW interpolation $(\mathbb{Y}_{\rho})_{\rho \in \Delta_3}$ between four input shapes $\mathbb{X}_1,\mathbb{X}_2,\mathbb{X}_3,\mathbb{X}_4$ shown in the corners.
• Figure 3: Pairwise GW distances (left) and confusion matrix (right) of the deformation dataset.
• Figure 4: Pairwise GW distances (left) and confusion matrix (right) of the Faust dataset.

Theorems & Definitions (34)

• Lemma 1
• Proof 1
• Lemma 2
• Proof 2
• Proposition 3
• Proof 3
• Lemma 1
• Proof 4
• Proposition 2
• Proof 5