Local stability and robustness of sparse dictionary learning in the presence of noise

TL;DR

This paper provides a theoretical analysis of sparse dictionary learning, demonstrating that local minima exist around the true dictionary even with noise and over-complete dictionaries, under a probabilistic model.

Contribution

It extends previous theoretical results to noisy, over-complete dictionaries and offers non-asymptotic bounds on key parameters affecting local minima.

Findings

Local minima exist around the true dictionary with high probability.

Analysis accounts for noise, over-completeness, and key problem parameters.

Provides bounds on how coherence and noise level scale with problem dimensions.

Abstract

A popular approach within the signal processing and machine learning communities consists in modelling signals as sparse linear combinations of atoms selected from a learned dictionary. While this paradigm has led to numerous empirical successes in various fields ranging from image to audio processing, there have only been a few theoretical arguments supporting these evidences. In particular, sparse coding, or sparse dictionary learning, relies on a non-convex procedure whose local minima have not been fully analyzed yet. In this paper, we consider a probabilistic model of sparse signals, and show that, with high probability, sparse coding admits a local minimum around the reference dictionary generating the signals. Our study takes into account the case of over-complete dictionaries and noisy signals, thus extending previous work limited to noiseless settings and/or under-complete dictionaries. The analysis we conduct is non-asymptotic and makes it possible to understand how the key quantities of the problem, such as the coherence or the level of noise, can scale with respect to the dimension of the signals, the number of atoms, the sparsity and the number of observations.