Distributional Reduction: Unifying Dimensionality Reduction and Clustering with Gromov-Wasserstein

TL;DR

This paper introduces distributional reduction, a unified framework based on Gromov-Wasserstein optimal transport, that combines dimensionality reduction and clustering into a single optimization approach, demonstrated on image and genomic data.

Contribution

It presents a novel unifying framework that integrates dimensionality reduction and clustering through optimal transport, specifically Gromov-Wasserstein, allowing joint analysis of data structure.

Findings

Effectively identifies low-dimensional prototypes across datasets

Unifies DR and clustering within a single optimization framework

Demonstrates applicability on image and genomic datasets

Abstract

Unsupervised learning aims to capture the underlying structure of potentially large and high-dimensional datasets. Traditionally, this involves using dimensionality reduction (DR) methods to project data onto lower-dimensional spaces or organizing points into meaningful clusters (clustering). In this work, we revisit these approaches under the lens of optimal transport and exhibit relationships with the Gromov-Wasserstein problem. This unveils a new general framework, called distributional reduction, that recovers DR and clustering as special cases and allows addressing them jointly within a single optimization problem. We empirically demonstrate its relevance to the identification of low-dimensional prototypes representing data at different scales, across multiple image and genomic datasets.