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

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

2011-02-04

arxiv.org/abs/1102.0844v1

Statistics

Machine Learning

14 pages, 9 figures. EE and JX were supported by NSF grants {DMS-0911277}, {PRISM-0948247}, MM by the German Academic Exchange

Scientific paper

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.

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