Local Geometry of Least Squares for Unmixing Signals with Parameter-Dependent Dictionaries

TL;DR

This paper develops a theoretical framework for least-squares unmixing of signals with parameter-dependent dictionaries, analyzing local convergence, stability, and the geometric interpretation of variable projection methods.

Contribution

It introduces the unmixing metric, provides stability and convergence guarantees, and offers a geometric perspective on variable projection for separable signal models.

Findings

Unmixing metric captures sensitivities of linear and nonlinear parameters.

Variable projection corresponds to restricting optimization to the manifold of optimal linear parameters.

Support separation controls convergence region size and recovery stability.

Abstract

Modeling signals as linear combinations of atoms from a dictionary is ubiquitous in modern signal processing. In the finite-dimensional setting, whenever atoms depend nonlinearly upon unknown parameters, the signal model is said to be separable. In this work, we study least-squares reconstruction of separable signals and establish a unified theoretical framework for their analysis. We introduce the unmixing metric, a distance that captures the distinct roles and sensitivities of linear and nonlinear parameters, and establish local convergence and stability guarantees under its topology. We then analyze variable projection from a geometric perspective, showing that it corresponds to restricting the optimization to the manifold of optimal linear parameters. This viewpoint provides a principled explanation for the improved algorithmic behavior of variable projection observed in practice, and produces sharp theoretical guarantees. The generic theory for separable problems is specialized to the case of point spread function (PSF) unmixing. We introduce a parametric notion of coherence and show that support separation directly controls both the size of the convergence region and the stability of recovery. Numerical experiments corroborate the theoretical predictions and demonstrate the practical relevance of the proposed framework.