A unified framework for non-negative matrix and tensor factorisations with a smoothed Wasserstein loss

TL;DR

This paper introduces a unified framework for non-negative matrix and tensor factorisations using a Wasserstein loss, capturing the geometric structure of data supported on metric spaces, with efficient computation and practical demonstrations.

Contribution

It presents a novel mathematical framework for non-negative factorisations with Wasserstein loss and develops an efficient convex dual-based computational method.

Findings

Effective factorisation of matrices and tensors with Wasserstein loss demonstrated

The approach captures data geometry better than traditional methods

Numerical experiments validate the method's applicability

Abstract

Non-negative matrix and tensor factorisations are a classical tool for finding low-dimensional representations of high-dimensional datasets. In applications such as imaging, datasets can be regarded as distributions supported on a space with metric structure. In such a setting, a loss function based on the Wasserstein distance of optimal transportation theory is a natural choice since it incorporates the underlying geometry of the data. We introduce a general mathematical framework for computing non-negative factorisations of both matrices and tensors with respect to an optimal transport loss. We derive an efficient computational method for its solution using a convex dual formulation, and demonstrate the applicability of this approach with several numerical illustrations with both matrix and tensor-valued data.