Geometry and Symmetry in Short-and-Sparse Deconvolution

TL;DR

This paper analyzes the geometric landscape of the short-and-sparse deconvolution problem, proposing a nonconvex optimization method that reliably recovers signals under certain conditions, accounting for intrinsic symmetry.

Contribution

It provides a geometric characterization of the optimization landscape for SaS deconvolution and introduces a provable nonconvex method for signal recovery.

Findings

Successful recovery with high probability under specified conditions.

Geometric landscape characterized on a union of subspaces.

Method accounts for intrinsic signed shift symmetry.

Abstract

We study the $\textit{Short-and-Sparse (SaS) deconvolution}$ problem of recovering a short signal $\mathbf a_0$ and a sparse signal $\mathbf x_0$ from their convolution. We propose a method based on nonconvex optimization, which under certain conditions recovers the target short and sparse signals, up to a signed shift symmetry which is intrinsic to this model. This symmetry plays a central role in shaping the optimization landscape for deconvolution. We give a $\textit{regional analysis}$, which characterizes this landscape geometrically, on a union of subspaces. Our geometric characterization holds when the length-$p_0$ short signal $\mathbf a_0$ has shift coherence $\mu$, and $\mathbf x_0$ follows a random sparsity model with sparsity rate $\theta \in \Bigl[\frac{c_1}{p_0}, \frac{c_2}{p_0\sqrt\mu + \sqrt{p_0}}\Bigr]\cdot\frac{1}{\log^2p_0}$. Based on this geometry, we give a provable method that successfully solves SaS deconvolution with high probability.