Sparse Linear Representation

TL;DR

This paper analyzes the effectiveness of sparse linear combinations for signal representation using overcomplete dictionaries, providing theoretical characterizations and an iterative method that achieves near-optimal results.

Contribution

It offers a precise characterization of optimal distortion based on dictionary size and introduces an iterative matching pursuit-like method with asymptotic optimality.

Findings

Signals can be sparsely represented with exponentially large dictionaries.

The iterative method approaches optimal representation asymptotically.

Provides a new proof of successive refinability for Gaussian sources.

Abstract

This paper studies the question of how well a signal can be reprsented by a sparse linear combination of reference signals from an overcomplete dictionary. When the dictionary size is exponential in the dimension of signal, then the exact characterization of the optimal distortion is given as a function of the dictionary size exponent and the number of reference signals for the linear representation. Roughly speaking, every signal is sparse if the dictionary size is exponentially large, no matter how small the exponent is. Furthermore, an iterative method similar to matching pursuit that successively finds the best reference signal at each stage gives asymptotically optimal representations. This method is essentially equivalent to successive refinement for multiple descriptions and provides a simple alternative proof of the successive refinability of white Gaussian sources.