A Convex Model for Nonnegative Matrix Factorization and Dimensionality Reduction on Physical Space

摘要

A collaborative convex framework for factoring a data matrix $X$ into a nonnegative 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 that 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 abundance identification and also show an application to blind source separation of nuclear magnetic resonance data.