Local Geometry of Least Squares for Unmixing Signals with Parameter-Dependent Dictionaries
For researchers working on signal unmixing with parameter-dependent dictionaries, this work provides a principled geometric understanding and sharp theoretical guarantees for variable projection, though it is primarily theoretical and incremental.
This paper develops a unified theoretical framework for least-squares reconstruction of separable signals, introducing the unmixing metric to capture the distinct roles of linear and nonlinear parameters and establishing local convergence and stability guarantees. The theory is specialized to point spread function unmixing, where a parametric coherence notion shows that support separation controls convergence region size and recovery stability.
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.