Local Geometry of Nonconvex Spike Deconvolution from Low-Pass Measurements

TL;DR

This paper analyzes the local geometry of nonconvex spike deconvolution from low-pass measurements and proposes preconditioned gradient descent methods, including an adaptive variant, to achieve efficient convergence under certain conditions.

Contribution

It introduces preconditioned gradient descent algorithms, with an adaptive version, that improve convergence rates for nonconvex spike deconvolution, especially when dealing with large amplitude variations.

Findings

Fixed preconditioner achieves linear convergence near ground truth with sufficient spike separation.

Convergence slows with large dynamic range of amplitudes.

Adaptive preconditioning accelerates convergence, independent of dynamic range.

Abstract

Spike deconvolution is the problem of recovering the point sources from their convolution with a known point spread function, which plays a fundamental role in many sensing and imaging applications. In this paper, we investigate the local geometry of recovering the parameters of point sources$\unicode{x2014}$including both amplitudes and locations$\unicode{x2014}$by minimizing a natural nonconvex least-squares loss function measuring the observation residuals. We propose preconditioned variants of gradient descent (GD), where the search direction is scaled via some carefully designed preconditioning matrices. We begin with a simple fixed preconditioner design, which adjusts the learning rates of the locations at a different scale from those of the amplitudes, and show it achieves a linear rate of convergence$\unicode{x2014}$in terms of entrywise errors$\unicode{x2014}$when initialized close to the ground truth, as long as the separation between the true spikes is sufficiently large. However, the convergence rate slows down significantly when the dynamic range of the source amplitudes is large. To bridge this issue, we introduce an adaptive preconditioner design, which compensates for the learning rates of different sources in an iteration-varying manner based on the current estimate. The adaptive design provably leads to an accelerated convergence rate that is independent of the dynamic range, highlighting the benefit of adaptive preconditioning in nonconvex spike deconvolution. Numerical experiments are provided to corroborate the theoretical findings.