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Let $A$ be an $n \times n$ matrix, $X$ be an $n \times p$ matrix and $Y =AX$. A challenging and important problem in data analysis, motivated bydictionary learning and other practical problems, is to recover both $A$ and$X$, given $Y$. Under normal circumstances, it is clear that this problem isunderdetermined. However, in the case when $X$ is sparse and random, Spielman,Wang and Wright showed that one can recover both $A$ and $X$ efficiently from$Y$ with high probability, given that $p$ (the number of samples) issufficiently large. Their method works for $p \ge C n^2 \log^ 2 n$ and theyconjectured that $p \ge C n \log n$ suffices. The bound $n \log n$ is sharp foran obvious information theoretical reason.  In this paper, we show that $p \ge C n \log^4 n$ suffices, matching theconjectural bound up to a polylogarithmic factor. The core of our proof is atheorem concerning $l_1$ concentration of random matrices, which is ofindependent interest.  Our proof of the concentration result is based on two ideas. The first is aneconomical way to apply the union bound. The second is a refined version ofBernstein's concentration inequality for the sum of independent variables. Bothhave nothing to do with random matrices and are applicable in general settings.

Dictionary Learning with Few Samples and Matrix Concentration