@article {473636, title = {Sparse Representation in Fourier and Local Bases Using ProSparse: A Probabilistic Analysis}, journal = {IEEE Transactions on Information Theory}, volume = {64}, number = {4}, year = {2018}, pages = {2639-2647}, 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.}, url = {https://arxiv.org/abs/1611.07971}, author = {Yue M. Lu and Jon O{\~n}ativia and Pier Luigi Dragotti} }