A convex model for non-negative matrix factorization and dimensionality reduction on physical space

TL;DR

This paper introduces a convex model for non-negative matrix factorization that ensures physically meaningful dictionaries and effective dimensionality reduction, with applications in hyperspectral imaging and blind source separation.

Contribution

The authors propose a novel convex framework for NMF with dictionary columns aligned to data, enabling exact relaxation of sparsity and improved interpretability.

Findings

Convex model guarantees physically meaningful dictionaries.

Exact relaxation of $l_0$ sparsity in noise-free data.

Effective application to hyperspectral endmember identification.

Abstract

A collaborative convex framework for factoring a data matrix $X$ into a non-negative product $AS$, with a sparse coefficient matrix $S$, is proposed. We restrict the columns of the dictionary matrix $A$ to coincide with certain columns of the data matrix $X$, thereby guaranteeing a physically meaningful dictionary and dimensionality reduction. We use $l_{1,\infty}$ regularization to select the dictionary from the data and show this leads to an exact convex relaxation of $l_0$ in the case of distinct noise free data. We also show how to relax the restriction-to-$X$ constraint by initializing an alternating minimization approach with the solution of the convex model, obtaining a dictionary close to but not necessarily in $X$. We focus on applications of the proposed framework to hyperspectral endmember and abundances identification and also show an application to blind source separation of NMR data.