Simple, Efficient, and Neural Algorithms for Sparse Coding

TL;DR

This paper introduces a unified framework for analyzing and designing simple, provably-guaranteed algorithms for sparse coding, achieving near-optimal recovery with improved efficiency and sample complexity.

Contribution

It provides the first efficient near-optimal sparse coding algorithm for incoherent dictionaries with provable guarantees and enhances understanding of alternating minimization heuristics.

Findings

Algorithms outperform existing methods in sparse recovery.

Achieves near information-theoretic limits efficiently.

Improves sample complexity over prior approaches.

Abstract

Sparse coding is a basic task in many fields including signal processing, neuroscience and machine learning where the goal is to learn a basis that enables a sparse representation of a given set of data, if one exists. Its standard formulation is as a non-convex optimization problem which is solved in practice by heuristics based on alternating minimization. Re- cent work has resulted in several algorithms for sparse coding with provable guarantees, but somewhat surprisingly these are outperformed by the simple alternating minimization heuristics. Here we give a general framework for understanding alternating minimization which we leverage to analyze existing heuristics and to design new ones also with provable guarantees. Some of these algorithms seem implementable on simple neural architectures, which was the original motivation of Olshausen and Field (1997a) in introducing sparse coding. We also give the first efficient algorithm for sparse coding that works almost up to the information theoretic limit for sparse recovery on incoherent dictionaries. All previous algorithms that approached or surpassed this limit run in time exponential in some natural parameter. Finally, our algorithms improve upon the sample complexity of existing approaches. We believe that our analysis framework will have applications in other settings where simple iterative algorithms are used.