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 $Y = AX$(uptoascalingandpermutationofcolumnsof$A$androwsof$X$). 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 $C$.