Blind Recovery of Sparse Signals from Subsampled Convolution

TL;DR

This paper introduces a new approach for blind deconvolution of sparse signals using a conic spectral flatness prior, providing an iterative algorithm with near-optimal sample complexity and performance guarantees, validated by numerical experiments.

Contribution

It proposes a novel spectral flatness prior combined with sparsity for blind deconvolution, along with an iterative algorithm that guarantees near-optimal recovery performance.

Findings

Algorithm achieves near-optimal sample complexity.

Empirical results match theoretical performance guarantees.

Spectral flatness prior improves identifiability in blind deconvolution.

Abstract

Subsampled blind deconvolution is the recovery of two unknown signals from samples of their convolution. To overcome the ill-posedness of this problem, solutions based on priors tailored to specific application have been developed in practical applications. In particular, sparsity models have provided promising priors. However, in spite of empirical success of these methods in many applications, existing analyses are rather limited in two main ways: by disparity between the theoretical assumptions on the signal and/or measurement model versus practical setups; or by failure to provide a performance guarantee for parameter values within the optimal regime defined by the information theoretic limits. In particular, it has been shown that a naive sparsity model is not a strong enough prior for identifiability in the blind deconvolution problem. Instead, in addition to sparsity, we adopt a conic constraint, which enforces spectral flatness of the signals. Under this prior, we provide an iterative algorithm that achieves guaranteed performance in blind deconvolution at near optimal sample complexity. Numerical results show the empirical performance of the iterative algorithm agrees with the performance guarantee.