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We consider the problem of recovering an invertible $n \times n$ matrix $A$and a sparse $n \times p$ random matrix $X$ based on the observation of $Y =AX$ (up to a scaling and permutation of columns of $A$ and rows of $X$). Usingonly elementary tools from the theory of empirical processes we show that aversion of the Er-SpUD algorithm by Spielman, Wang and Wright with highprobability recovers $A$ and $X$ exactly, provided that $p \ge Cn\log n$, whichis optimal up to the constant $C$.

A note on the sample complexity of the Er-SpUD algorithm by Spielman, Wang and Wright for exact recovery of sparsely used dictionaries