Towards Optimal Transport with Global Invariances

TL;DR

This paper introduces a framework for optimal transport that accounts for global invariances like rotation and reflection, enabling better alignment of learned representations such as word embeddings.

Contribution

It formulates a joint optimization approach for transport and invariances, providing algorithms and demonstrating effectiveness on tasks like unsupervised word translation.

Findings

Effective handling of global invariances improves transport alignment.

Algorithms successfully optimize transport and transformations jointly.

Promising results on unsupervised word translation benchmark.

Abstract

Many problems in machine learning involve calculating correspondences between sets of objects, such as point clouds or images. Discrete optimal transport provides a natural and successful approach to such tasks whenever the two sets of objects can be represented in the same space, or at least distances between them can be directly evaluated. Unfortunately neither requirement is likely to hold when object representations are learned from data. Indeed, automatically derived representations such as word embeddings are typically fixed only up to some global transformations, for example, reflection or rotation. As a result, pairwise distances across two such instances are ill-defined without specifying their relative transformation. In this work, we propose a general framework for optimal transport in the presence of latent global transformations. We cast the problem as a joint optimization over transport couplings and transformations chosen from a flexible class of invariances, propose algorithms to solve it, and show promising results in various tasks, including a popular unsupervised word translation benchmark.