Scalable Optimal Transport in High Dimensions for Graph Distances, Embedding Alignment, and More

TL;DR

This paper introduces log-linear time approximations for optimal transport in high-dimensional spaces, enabling faster and more accurate applications in embedding alignment and graph distance regression.

Contribution

It proposes LSH-based sparse and Nyström-based locally corrected approximations for OT, improving scalability and accuracy in high-dimensional deep learning tasks.

Findings

Speeds up word embedding alignment by 3x and improves accuracy by 3.1 percentage points.

Proposes Graph Transport Network (GTN) that outperforms previous models by 48%.

Achieves log-linear scaling in graph distance regression.

Abstract

The current best practice for computing optimal transport (OT) is via entropy regularization and Sinkhorn iterations. This algorithm runs in quadratic time as it requires the full pairwise cost matrix, which is prohibitively expensive for large sets of objects. In this work we propose two effective log-linear time approximations of the cost matrix: First, a sparse approximation based on locality-sensitive hashing (LSH) and, second, a Nystr\"om approximation with LSH-based sparse corrections, which we call locally corrected Nystr\"om (LCN). These approximations enable general log-linear time algorithms for entropy-regularized OT that perform well even for the complex, high-dimensional spaces common in deep learning. We analyse these approximations theoretically and evaluate them experimentally both directly and end-to-end as a component for real-world applications. Using our approximations for unsupervised word embedding alignment enables us to speed up a state-of-the-art method by a factor of 3 while also improving the accuracy by 3.1 percentage points without any additional model changes. For graph distance regression we propose the graph transport network (GTN), which combines graph neural networks (GNNs) with enhanced Sinkhorn. GTN outcompetes previous models by 48% and still scales log-linearly in the number of nodes.