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

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, ∞}
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.