The Gromov-Wasserstein (GW) distance is an extension of the optimal transport problem that allows one to match objects between incomparable spaces. At its core, the GW distance is specified as the solution of a non-convex quadratic program and is not known to be tractable to solve. In particular, existing solvers for the GW distance are only able to find locally optimal solutions. In this work, we propose a semi-definite programming (SDP) relaxation of the GW distance. The relaxation can be viewed as the Lagrangian dual of the GW distance augmented with constraints that relate to the linear and quadratic terms of transportation plans. In particular, our relaxation provides a tractable (polynomial-time) algorithm to compute globally optimal transportation plans (in some instances) together with an accompanying proof of global optimality. Our numerical experiments suggest that the proposed relaxation is strong in that it frequently computes the globally optimal solution. Our Python implementation is available at https://github.com/tbng/gwsdp.

The Gromov-Wasserstein (GW) distance is an extension of the optimal transport problem that allows one to match objects between incomparable spaces. At its core, the GW distance is specified as the solution of a non-convex quadratic program and is not known to be tractable to solve. In particular, existing solvers for the GW distance are only able to find locally optimal solutions. In this work, we propose a semi-definite programming (SDP) relaxation of the GW distance. The relaxation can be viewed as the Lagrangian dual of the GW distance augmented with constraints that relate to the linear and quadratic terms of transportation plans. In particular, our relaxation provides a tractable (polynomial-time) algorithm to compute globally optimal transportation plans (in some instances) together with an accompanying proof of global optimality. Our numerical experiments suggest that the proposed relaxation is strong in that it frequently computes the globally optimal solution. Our Python implementation is available at https://github.com/tbng/gwsdp.

Investigating Fine-and Coarse-grained Structural Correspondences Between Deep Neural Networks and Human Object Image Similarity Judgments Using Unsupervised Alignment Soh Takahashi, Masaru Sasaki, Ken Takeda, Masafumi Oizumi Introduces an unsupervised alignment to assess human-like object representations of DNNs. CLIP models show highest fine-grained matching (20% top-1 match). This underscores the role of linguistic cues in refining representations. Image-only self-supervised models lack fine matching with human representations. Rather, they capture coarse category structures, hinting at prelinguistic links. Abstract The learning mechanisms by which humans acquire internal representations of objects are not fully understood. Deep neural networks (DNNs) have emerged as a useful tool for investigating this question, as they have internal representations similar to those of humans as a byproduct of optimizing their objective functions. While previous studies have shown that models trained with various learning paradigms--such as supervised, self-supervised, and CLIP--acquire human-like representations, it remains unclear whether their similarity to human representations is primarily at a coarse category level or extends to finer details. Here, we employ an unsupervised alignment method based on Gromov-Wasserstein Optimal Transport to compare human and model object representations at both fine-grained and coarse-grained levels. The unique feature of this method compared to conventional representational similarity analysis is that it estimates optimal fine-grained mappings between the representation of each object in human and model representations. We used this unsupervised alignment method to assess the extent to which the representation of each object in humans is correctly mapped to the corresponding representation of the same object in models. Using human similarity judgments of 1,854 objects from the THINGS dataset, we find that models trained with CLIP consistently achieve strong fine-and coarse-grained matching with human object representations.

The Gromov-Wasserstein (GW) distance is an extension of the optimal transport problem that allows one to match objects between incomparable spaces. At its core, the GW distance is specified as the solution of a non-convex quadratic program and is not known to be tractable to solve. In particular, existing solvers for the GW distance are only able to find locally optimal solutions. In this work, we propose a semi-definite programming (SDP) relaxation of the GW distance. The relaxation can be viewed as the Lagrangian dual of the GW distance augmented with constraints that relate to the linear and quadratic terms of transportation plans. In particular, our relaxation provides a tractable (polynomial-time) algorithm to compute globally optimal transportation plans (in some instances) together with an accompanying proof of global optimality. Our numerical experiments suggest that the proposed relaxation is strong in that it frequently computes the globally optimal solution. Our Python implementation is available at https://github.com/tbng/gwsdp.

On the other hand, the numerical resolution of OT problems using entropic regularization has given rise to many applications, but because there are no known closed-form solutions for entropic regularized OT problems, these approaches are mostly algorithmic, not informed by elegant closed forms.

On the other hand, the numerical resolution of OT problems using entropic regularization has given rise to many applications, but because there are no known closed-form solutions for entropic regularized OT problems, these approaches are mostly algorithmic, not informed by elegant closed forms.