Dictionary Learning with Few Samples and Matrix Concentration

TL;DR

This paper improves the sample complexity bound for recovering a sparse matrix factorization, showing that near-optimal sample size suffices using advanced concentration inequalities and union bound techniques.

Contribution

It proves that $p \,\geq\, C n \log^4 n$ samples are enough for recovery, approaching the conjectured optimal bound up to a polylogarithmic factor.

Findings

Achieved near-optimal sample complexity for dictionary learning.

Developed new concentration inequalities for random matrices.

Introduced refined Bernstein's inequality and union bound techniques.

Abstract

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 by dictionary learning and other practical problems, is to recover both $A$ and $X$, given $Y$. Under normal circumstances, it is clear that this problem is underdetermined. 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) is sufficiently large. Their method works for $p \ge C n^2 \log^ 2 n$ and they conjectured that $p \ge C n \log n$ suffices. The bound $n \log n$ is sharp for an obvious information theoretical reason. In this paper, we show that $p \ge C n \log^4 n$ suffices, matching the conjectural bound up to a polylogarithmic factor. The core of our proof is a theorem concerning $l_1$ concentration of random matrices, which is of independent interest. Our proof of the concentration result is based on two ideas. The first is an economical way to apply the union bound. The second is a refined version of Bernstein's concentration inequality for the sum of independent variables. Both have nothing to do with random matrices and are applicable in general settings.