Sparse Representation in Fourier and Local Bases Using ProSparse: A Probabilistic Analysis

TL;DR

This paper provides a probabilistic analysis of ProSparse, an algorithm for sparse signal representation in overcomplete dictionaries, revealing success probabilities and phase transition behavior in high-dimensional settings.

Contribution

It introduces a probabilistic average-case analysis of ProSparse, including exact success probabilities and asymptotic phase transition characterization.

Findings

Exact success probabilities derived for ProSparse

Identification of a sharp phase transition in high dimensions

Analysis based on generating functions

Abstract

Finding the sparse representation of a signal in an overcomplete dictionary has attracted a lot of attention over the past years. This paper studies ProSparse, a new polynomial complexity algorithm that solves the sparse representation problem when the underlying dictionary is the union of a Vandermonde matrix and a banded matrix. Unlike our previous work which establishes deterministic (worst-case) sparsity bounds for ProSparse to succeed, this paper presents a probabilistic average-case analysis of the algorithm. Based on a generating-function approach, closed-form expressions for the exact success probabilities of ProSparse are given. The success probabilities are also analyzed in the high-dimensional regime. This asymptotic analysis characterizes a sharp phase transition phenomenon regarding the performance of the algorithm.