The Joint Gromov Wasserstein Objective for Multiple Object Matching

TL;DR

The paper proposes a joint extension of the Gromov-Wasserstein distance, termed Joint Gromov-Wasserstein ($\mathcal{JGW}_p$), to match collections of mm-spaces and address multiple-to-multiple partial matching. It provides theoretical guarantees linking zero JGW to partial isomorphism and proves convergence under random point sampling. A discretized, entropic-regularized optimization framework is derived, employing a KL-based update driven by a matrix operator $\Lambda(\mu)$ and entropy $H(\mu)$. Empirical results across partial matching, 2D/3D shape matching, and cryo-EM density map alignment demonstrate superior accuracy and efficiency relative to unbalanced/partial GW variants, highlighting potential impacts in computer graphics and structural biology.

Abstract

The Gromov-Wasserstein (GW) distance serves as a powerful tool for matching objects in metric spaces. However, its traditional formulation is constrained to pairwise matching between single objects, limiting its utility in scenarios and applications requiring multiple-to-one or multiple-to-multiple object matching. In this paper, we introduce the Joint Gromov-Wasserstein (JGW) objective and extend the original framework of GW to enable simultaneous matching between collections of objects. Our formulation provides a non-negative dissimilarity measure that identifies partially isomorphic distributions of mm-spaces, with point sampling convergence. We also show that the objective can be formulated and solved for point cloud object representations by adapting traditional algorithms in Optimal Transport, including entropic regularization. Our benchmarking with other variants of GW for partial matching indicates superior performance in accuracy and computational efficiency of our method, while experiments on both synthetic and real-world datasets show its effectiveness for multiple shape matching, including geometric shapes and biomolecular complexes, suggesting promising applications for solving complex matching problems across diverse domains, including computer graphics and structural biology.

Figures (5)

• Figure 1: a. A simple example of a discrete mm-space with values of $d_x$ and $\mu_x$ provided. b. An example of a discrete distribution of mm-spaces containing two clusters with values of $d_x$, $\mu_x$, $s_x$ provided. Each point's size corresponds to the value of $\mu$ at that point.
• Figure 2: Performance comparison of GW variants for partial matching. We evaluate mPGWchapel2020partial, PGWbai2024efficient, UGWpeyre2019computational, and our proposed JGW approach. a. Source distribution (blue) comprising $200$ points sampled from an Archimedean spiral, and target distribution containing $200$ points from the same spiral plus $100$ noise points from a standard normal distribution (red). b. Couplings computed by each method, demonstrating JGW's superior performance in handling partial matches.
• Figure 3: Comparison of the quality of the couplings generated by UGW and JGW on the same example as Figure \ref{['fig:jgw-partial-coupling']}. a. Couplings computed by each method, with visualization of how a single source point (the leftmost point in the source) is matched across the target distribution (purple edges). Both UGW and JGW distribute mass across multiple target points due to regularization, with JGW achieving lower variance. b. Violin plot showing the variance of coupled target points for each source point, confirming JGW's better mass concentration.
• Figure 4: Performance of JGW in matching shapes involving 2D and 3D data. a. The source and target distributions created using different typesets and combinations of letters "A", "B", and "C". Performance of JGW in matching the source space and target, each color shows the clusters of the coupled most points to a point of the target distribution. b. The source and target space created from 3D meshes of human body for CAPOD dataset papadakis2014canonically. The results of the 3D experiments is demonstrated in the same way as before. This diagram shows the perfect performance of this method in matching the hands and the body, while mismatching some parts of the legs. c. The source and target space created from an example of SHREC'16 dataset cosmo2016shrec. The results of the 3D experiments is demonstrated in the same way as before. This diagram shows the near-perfect performance of this method. To quantitatively evaluate the mapping quality, we employed a standard measure introduced in kim2011blended: the geodesic distance between ground truth and computed corresponding points, normalized by the square root of the full shape's area, and illustrated a cumulative distribution function (CDF) of this measure across all mesh vertices.
• Figure 5: Performance of JGW on matching biomolecular complexes compared to riahi2025alignment. a. We used the atomic structure of PDB:1I3Q cramer2001structural and simplified it into 3 chains. Then applied JGW and sequential partial matching with UGW riahi2025alignmentto reconstruct it by aligning its chains into the whole map. b. The results of alignment of each chain using UGW and one-by-one alignment of chains and JGW (ours). In each diagram, the blue structure shows the ground truth while the red one represents the aligned one.

Theorems & Definitions (22)

• Definition 1
• Definition 2
• Definition 3
• Definition 4
• Theorem 5: memoli2011gromov
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• Theorem 9
• Definition 10