Sparse Separable Nonnegative Matrix Factorization

TL;DR

This paper introduces a new variant of nonnegative matrix factorization called sparse separable NMF (SSNMF), designed for blind source separation tasks like multispectral image unmixing, and provides an algorithm with theoretical recovery guarantees.

Contribution

The paper proposes SSNMF, combining separability and sparsity, proves its NP-completeness, and develops an algorithm with proven source recovery in noiseless conditions.

Findings

Algorithm successfully recovers sources in synthetic data.

Effective for multispectral image unmixing.

Proven NP-completeness of SSNMF.

Abstract

We propose a new variant of nonnegative matrix factorization (NMF), combining separability and sparsity assumptions. Separability requires that the columns of the first NMF factor are equal to columns of the input matrix, while sparsity requires that the columns of the second NMF factor are sparse. We call this variant sparse separable NMF (SSNMF), which we prove to be NP-complete, as opposed to separable NMF which can be solved in polynomial time. The main motivation to consider this new model is to handle underdetermined blind source separation problems, such as multispectral image unmixing. We introduce an algorithm to solve SSNMF, based on the successive nonnegative projection algorithm (SNPA, an effective algorithm for separable NMF), and an exact sparse nonnegative least squares solver. We prove that, in noiseless settings and under mild assumptions, our algorithm recovers the true underlying sources. This is illustrated by experiments on synthetic data sets and the unmixing of a multispectral image.