Matrix-wise $\ell_0$-constrained Sparse Nonnegative Least Squares

TL;DR

This paper introduces a novel matrix-wise sparsity constraint for nonnegative least squares problems with multiple right-hand sides, improving interpretability and noise reduction in applications like image analysis.

Contribution

It proposes a new formulation for matrix-wise sparsity in MNNLS and a two-step algorithm that optimizes solutions along Pareto fronts for better accuracy.

Findings

Outperforms state-of-the-art sparse coding heuristics

Provides more accurate solutions in facial and hyperspectral image experiments

Effectively balances reconstruction error and sparsity

Abstract

Nonnegative least squares problems with multiple right-hand sides (MNNLS) arise in models that rely on additive linear combinations. In particular, they are at the core of most nonnegative matrix factorization algorithms and have many applications. The nonnegativity constraint is known to naturally favor sparsity, that is, solutions with few non-zero entries. However, it is often useful to further enhance this sparsity, as it improves the interpretability of the results and helps reducing noise, which leads to the sparse MNNLS problem. In this paper, as opposed to most previous works that enforce sparsity column- or row-wise, we first introduce a novel formulation for sparse MNNLS, with a matrix-wise sparsity constraint. Then, we present a two-step algorithm to tackle this problem. The first step divides sparse MNNLS in subproblems, one per column of the original problem. It then uses different algorithms to produce, either exactly or approximately, a Pareto front for each subproblem, that is, to produce a set of solutions representing different tradeoffs between reconstruction error and sparsity. The second step selects solutions among these Pareto fronts in order to build a sparsity-constrained matrix that minimizes the reconstruction error. We perform experiments on facial and hyperspectral images, and we show that our proposed two-step approach provides more accurate results than state-of-the-art sparse coding heuristics applied both column-wise and globally.