  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.
