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

The problem of recovering a mixture of spike signals convolved with distinct kernels (point spread functions, PSFs) lying on a parametric manifold, under the assumption that the spike locations are known, is studied. We formulate the projected non-linear least squares estimator for kernel unmixing and establish a lower bound on the radius of the region of strong convexity 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 role of the speed of decay of the PSF class in the problem’s conditioning and confirm theoretical findings. Finally, we deploy the estimator on real-world laser-induced breakdown spectroscopy (LIBS) data, removing the need for manual calibration and demonstrating practical relevance.