Emergent Mind
A note on the sample complexity of the Er-SpUD algorithm by Spielman, Wang and Wright for exact recovery of sparsely used dictionaries
(1601.02049)
Published Jan 8, 2016
in
math.PR
,
cs.LG
,
math.ST
,
and
stat.TH
Abstract
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 $C$.
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