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

TL;DR

This work tackles unmixing of spike-like signals convolved with multiple PSFs that lie on a parameterized manifold, assuming known spike supports. It formulates a non-linear least-squares estimator for the amplitudes and PSF-shape parameters and derives a noiseless lower bound on the radius of the strong basin of attraction, expressed via new coherence and interference functions tied to the PSF manifold and minimum segment separation $\Delta$. The main result guarantees convergence of line-search methods when initialized inside a quantified basin, with bounds depending on constants $c_-^{\star}$, $c_+^{\star}$, $r^{\star}$, and $q^{\star}$. Numerical experiments on synthetic Lorentz PSFs and LIBS data validate the theory and demonstrate practical applicability to spectral analysis, including aluminum concentration estimation. The methodology provides a pathway towards scalable, non-convex unmixing in settings where PSFs are governed by a manifold and contribute to accurate, data-driven spectroscopy interpretation.

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.

Figures (4)

• Figure 1: Graphs of the coherence functions (left) and the interference functions (right) for the class of Lorentz PSF $g(\theta, t) = \theta / (\pi (\theta^2 + t^2))$ in the range $\Delta \in [0.1, 2]$ for the parameters $\theta_1 = 0.2$ and $\theta_2=0.1$.
• Figure 2: The level sets of the restricted map $\bm{\theta} \mapsto \mathcal{L}(\bm{\theta}, \bm{\eta}^\star)$, and the strong basin radii (in white/dashed) predicted by Theorem \ref{['thm:main-result']}, for decreasing separation parameter $\Delta$.
• Figure 3: Success rate of scipy's least squares solver to recover the ground truth $\bm{\theta}^\star$ as a function of of the $\ell_{\infty}$-distance between the initial guess $\bm{\theta}^{0}$ and the ground truth $\bm{\theta}^{\star}$. The results are averaged over 50 trials. The lower bound on the radius of the basin predicted by Theorem \ref{['thm:main-result']} is shown in a dashed vertical line.
• Figure 4: Sampled and fitted LIBS spectrum of an aluminum compound with corresponding ions indicated alongside their estimated concentrations.

Theorems & Definitions (3)

• Theorem 1
• Lemma 2
• Lemma 3