Sparse and spurious: dictionary learning with noise and outliers

TL;DR

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

Contribution

It extends previous theoretical results to noisy, outlier-prone, and over-complete settings, offering non-asymptotic guarantees for sparse coding.

Findings

Local minima exist around the true dictionary with high probability.

Analysis accounts for noise, outliers, and over-complete dictionaries.

Key quantities like coherence and noise level scale with signal dimension and data size.

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, noisy signals, and possible outliers, 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.