Convergence Guarantees for Unmixing PSFs over a Manifold with Non-Convex Optimization

TL;DR

This paper establishes convergence guarantees for a non-linear least squares method to recover mixture parameters of spike signals convolved with PSFs on a manifold, with theoretical bounds and practical validation.

Contribution

It introduces coherence and interference functions to analyze the conditioning of PSF manifolds and derives convergence bounds for the non-linear least squares estimator.

Findings

Theoretical lower bounds on the convergence region are derived.

Numerical experiments validate the theoretical bounds.

The method is effective on spectral data from laser-induced breakdown spectroscopy.

Abstract

The problem of recovering the parameters of a mixture of spike signals convolved with different PSFs is considered. Herein, the spike support is assumed to be known, while the PSFs lie on a manifold. A non-linear least squares estimator of the mixture parameters is formulated. In the absence of noise, a lower bound on the radius of the strong basin of attraction i.e., the region of convergence, is derived. Key to the analysis is the introduction of coherence and interference functions, which capture the conditioning of the PSF manifold in terms of the minimal separation of the support. Numerical experiments validate the theoretical findings. Finally, the practicality and efficacy of the non-linear least squares approach are showcased on spectral data from laser-induced breakdown spectroscopy.