title: A convex model for nonnegative matrix factorization and dimensionality reduction on physical space.
abstract: 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.
authors.affiliation: Department of Mathematics, University of California at Irvine, Irvine, CA 92697-3875, USA. eesser@math.uci.edu
authors.forename: Ernie
authors.initials: E
authors.lastname: Esser
authors.forename: Michael
authors.initials: M
authors.lastname: Möller
authors.forename: Stanley
authors.initials: S
authors.lastname: Osher
authors.forename: Guillermo
authors.initials: G
authors.lastname: Sapiro
authors.forename: Jack
authors.initials: J
authors.lastname: Xin