Strong Basin of Attraction for Unmixing Kernels With the Variable Projection Method

The problem of recovering a mixture of spike signals convolved with distinct point spread functions (PSFs) lying on a parametric manifold, under the assumption that the spike locations are known, is studied. The PSF unmixing problem is formulated as a projected non-linear least squares estimator. A lower bound on the radius of the region of strong convexity is established in the presence of noise as a function of the manifold coherence and Lipschitz properties, guaranteeing convergence and stability of the optimization program. Numerical experiments highlight the speed of decay of the PSF class in the problem's conditioning and confirm theoretical findings. Finally, the proposed estimator is deployed on real-world spectroscopic data from laser-induced breakdown spectroscopy (LIBS), removing the need for manual calibration and validating the method's practical relevance.