## A Note on the Sample Complexity of the Er-SpUD Algorithm by Spielman, Wang and Wright for Exact Recovery of Sparsely Used Dictionaries

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 $X$). Using only elementary tools from the theory of empirical processes we show that a version of the Er-SpUD algorithm by Spielman, Wang and Wright with high probability recovers $A$ and $X$ exactly, provided that $p \ge Cn\log n$, which is optimal up to the constant $C$.