Preconditioned Gradient Descent for Spike Deconvolution and Sketched Mixture Learning
Abstract
A unified framework for analyzing the local geometry of two fundamental point-source recovery problems: spike deconvolution and mixture models (MM) from sketches, is proposed. The framework consists of formulating the recovery of the model parameters, describing the amplitudes (or weights) and locations of the point sources to reconstruct, as a non-linear least squares problem. The local geometry of the non-convex objective function is characterized in terms of the shape of the point-spread function involved in the signal acquisition, the minimal separation between the point sources, the dynamic range of the amplitudes, and the signal-to-noise ratio. Then, the basin of attraction for the preconditioned gradient descent algorithm (PGD), a scalable firstorder optimization method, towards the right local minimizer is studied, establishing provable statistical guarantees of the method under the proviso of a good initialization. In particular, while incurring no computational overhead compared to vanilla gradient descent, PGD is shown to converge linearly towards the global minimizer of the loss. PGD analysis is also extended to online settings, which is particularly relevant when samples are acquired incrementally. Finally, numerical simulations are conducted to validate the theoretical findings and to showcase the method's effectiveness.
Cite
@misc{gabet:hal-05722525,
author = {Gabet, Joseph and Ferreira Da Costa, Maxime},
title = {Preconditioned Gradient Descent for Spike Deconvolution and Sketched Mixture Learning},
year = {2026},
eprint = {hal-05722525},
eprinttype = {hal}
}