Towards Tuning-Free Minimum-Volume Nonnegative Matrix Factorization

TL;DR

This paper introduces a tuning-free minimum-volume NMF formulation inspired by the square-root lasso, along with a convergent algorithm, demonstrating robustness to noise level variations in data.

Contribution

It proposes a novel tuning-free minimum-volume NMF model and an associated MM algorithm with global convergence guarantees.

Findings

The method's optimal tuning parameter is insensitive to noise level.

The proposed algorithm converges globally.

Empirical results show robustness to noise variations.

Abstract

Nonnegative Matrix Factorization (NMF) is a versatile and powerful tool for discovering latent structures in data matrices, with many variations proposed in the literature. Recently, Leplat et al.\@ (2019) introduced a minimum-volume NMF for the identifiable recovery of rank-deficient matrices in the presence of noise. The performance of their formulation, however, requires the selection of a tuning parameter whose optimal value depends on the unknown noise level. In this work, we propose an alternative formulation of minimum-volume NMF inspired by the square-root lasso and its tuning-free properties. Our formulation also requires the selection of a tuning parameter, but its optimal value does not depend on the noise level. To fit our NMF model, we propose a majorization-minimization (MM) algorithm that comes with global convergence guarantees. We show empirically that the optimal choice of our tuning parameter is insensitive to the noise level in the data.