Global Convergence of ESPRIT with Preconditioned First-Order Methods for Spike Deconvolution

TL;DR

This paper introduces a novel method combining ESPRIT with preconditioned gradient descent for spike deconvolution, achieving super-linear convergence and robustness in recovering point sources from convoluted signals.

Contribution

It presents a new approach that integrates ESPRIT with adaptively preconditioned gradient descent, providing global convergence guarantees in 1D spike deconvolution.

Findings

Proven super-linear convergence rate for the proposed method

Global convergence guarantees in one-dimensional settings

Numerical simulations confirm robustness and effectiveness

Abstract

Spike deconvolution is the problem of recovering point sources from their convolution with a known point spread function, playing a fundamental role in many sensing and imaging applications. This paper proposes a novel approach combining ESPRIT with Preconditioned Gradient Descent (PGD) to estimate the amplitudes and locations of the point sources by a non-linear least squares. The preconditioning matrices are adaptively designed to account for variations in the learning process, ensuring a proven super-linear convergence rate. We provide local convergence guarantees for PGD and performance analysis of ESPRIT reconstruction, leading to global convergence guarantees for our method in one-dimensional settings with multiple snapshots, demonstrating its robustness and effectiveness. Numerical simulations corroborate the performance of the proposed approach for spike deconvolution.