Journal article

### A convex model for nonnegative matrix factorization and dimensionality reduction on physical space

Esser E, M??ller M, Osher S, Sapiro G, Xin J ...see all

IEEE Transactions on Image Processing, vol. 21, issue 7 (2012) pp. 3239-3252

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#### 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.

#### Author-supplied keywords

• Blind source separation (BSS)
• dictionary learning
• dimensionality reduction
• hyperspectral endmember detection
• nonnegative matrix factorization (NMF)
• subset selection

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