Maximum-Volume Nonnegative Matrix Factorization

TL;DR

This paper introduces maximum-volume NMF, a novel approach that enhances interpretability and sparsity in data decomposition, especially under noisy conditions, and demonstrates its effectiveness in hyperspectral unmixing.

Contribution

It proposes the MaxVol NMF method, analyzes its properties, and develops algorithms, offering a new perspective compared to traditional MinVol NMF.

Findings

MaxVol NMF is more effective for sparse decomposition.

Solutions of MaxVol NMF correspond to clustering columns in disjoint groups.

Normalized MaxVol NMF performs better and bridges standard and orthogonal NMF.

Abstract

Nonnegative matrix factorization (NMF) is a popular data embedding technique. Given a nonnegative data matrix $X$, it aims at finding two lower dimensional matrices, $W$ and $H$, such that $X\approx WH$, where the factors $W$ and $H$ are constrained to be element-wise nonnegative. The factor $W$ serves as a basis for the columns of $X$. In order to obtain more interpretable and unique solutions, minimum-volume NMF (MinVol NMF) minimizes the volume of $W$. In this paper, we consider the dual approach, where the volume of $H$ is maximized instead; this is referred to as maximum-volume NMF (MaxVol NMF). MaxVol NMF is identifiable under the same conditions as MinVol NMF in the noiseless case, but it behaves rather differently in the presence of noise. In practice, MaxVol NMF is much more effective to extract a sparse decomposition and does not generate rank-deficient solutions. In fact, we prove that the solutions of MaxVol NMF with the largest volume correspond to clustering the columns of $X$ in disjoint clusters, while the solutions of MinVol NMF with smallest volume are rank deficient. We propose two algorithms to solve MaxVol NMF. We also present a normalized variant of MaxVol NMF that exhibits better performance than MinVol NMF and MaxVol NMF, and can be interpreted as a continuum between standard NMF and orthogonal NMF. We illustrate our results in the context of hyperspectral unmixing.