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A Riemannian Approach to Ground Metric Learning for Optimal Transport

Pratik Jawanpuria, Dai Shi, Bamdev Mishra, Junbin Gao

TL;DR

The paper addresses how the ground metric used in optimal transport can be learned directly from data by parameterizing it with a symmetric positive definite matrix $\mathbf A$. By exploiting the Riemannian geometry of the SPD manifold, the authors derive a closed-form update for $\mathbf A$ and a Sinkhorn-based update for the transport plan $\gamma$, enabling an efficient alternating optimization. The objective can be interpreted as the geodesic midpoint (geometric mean) between $\mathbf C_{\gamma}^{-1}$ and a fixed matrix $\mathbf D$ on the SPD manifold, providing a principled regularization. Experiments on MNIST and Caltech-Office demonstrate that jointly learning the ground metric and transport plan improves domain adaptation performance over fixed-metric OT baselines, validating the practical impact of data-driven ground metric learning in OT.

Abstract

Optimal transport (OT) theory has attracted much attention in machine learning and signal processing applications. OT defines a notion of distance between probability distributions of source and target data points. A crucial factor that influences OT-based distances is the ground metric of the embedding space in which the source and target data points lie. In this work, we propose to learn a suitable latent ground metric parameterized by a symmetric positive definite matrix. We use the rich Riemannian geometry of symmetric positive definite matrices to jointly learn the OT distance along with the ground metric. Empirical results illustrate the efficacy of the learned metric in OT-based domain adaptation.

A Riemannian Approach to Ground Metric Learning for Optimal Transport

TL;DR

The paper addresses how the ground metric used in optimal transport can be learned directly from data by parameterizing it with a symmetric positive definite matrix . By exploiting the Riemannian geometry of the SPD manifold, the authors derive a closed-form update for and a Sinkhorn-based update for the transport plan , enabling an efficient alternating optimization. The objective can be interpreted as the geodesic midpoint (geometric mean) between and a fixed matrix on the SPD manifold, providing a principled regularization. Experiments on MNIST and Caltech-Office demonstrate that jointly learning the ground metric and transport plan improves domain adaptation performance over fixed-metric OT baselines, validating the practical impact of data-driven ground metric learning in OT.

Abstract

Optimal transport (OT) theory has attracted much attention in machine learning and signal processing applications. OT defines a notion of distance between probability distributions of source and target data points. A crucial factor that influences OT-based distances is the ground metric of the embedding space in which the source and target data points lie. In this work, we propose to learn a suitable latent ground metric parameterized by a symmetric positive definite matrix. We use the rich Riemannian geometry of symmetric positive definite matrices to jointly learn the OT distance along with the ground metric. Empirical results illustrate the efficacy of the learned metric in OT-based domain adaptation.
Paper Structure (11 sections, 1 theorem, 9 equations, 2 tables, 1 algorithm)

This paper contains 11 sections, 1 theorem, 9 equations, 2 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Background and related works
  3. Proposed approach
  4. Optimization algorithm
  5. Metric learning problem: fixing $\gamma$, solve (\ref{['eq:reg_mahalanobis_OT']}) for $\mathbf A$
  6. OT problem: fixing $\mathbf A$, solve (\ref{['eq:reg_mahalanobis_OT']}) for $\gamma$
  7. Experiments
  8. Barycentric projection for domain adaptation
  9. Experimental setup
  10. Results and discussion
  11. Conclusion

Key Result

Proposition 4.1

Given $\gamma$, the global optimal $\mathbf A^*$ for Problem (eq:fix_gamma) is: where $\mathbf C_\gamma = \sum_i\sum_j \gamma_{ij}({\mathbf x}_i-{\mathbf z}_j)({\mathbf x}_i-{\mathbf z}_j)^\top$.

Theorems & Definitions (1)

  • Proposition 4.1