# Characterization of the Basin of Convexity for Multi-Snapshot Spike Deconvolution via Variable Projection

arXiv: https://arxiv.org/abs/2607.09593 · HTML: https://mlsanity.com/paper/2607.09593/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 8 of its 47 references that are in the map.
- First posted on arXiv 10 Jul 2026.

## Builds on (most cited first, 8 of 8)

- [Superresolution without Separation](https://mlsanity.com/paper/1506.03144/) (2015)
- [Channel State Information-Free Location-Privacy Enhancement: Fake Path Injection](https://mlsanity.com/paper/2307.05442/) (2023)
- [Harnessing Sparsity over the Continuum: Atomic Norm Minimization for Super Resolution](https://mlsanity.com/paper/1904.04283/) (2019)
- [Stability and Super-resolution of MUSIC and ESPRIT for Multi-snapshot Spectral Estimation](https://mlsanity.com/paper/2105.14304/) (2021)
- [Nonasymptotic performance analysis of ESPRIT and spatial-smoothing ESPRIT](https://mlsanity.com/paper/2201.03457/) (2022)
- [On the Stability of Super-Resolution and a Beurling-Selberg Type Extremal Problem](https://mlsanity.com/paper/2202.02932/) (2022)
- [Local Geometry of Nonconvex Spike Deconvolution from Low-Pass Measurements](https://mlsanity.com/paper/2208.10073/) (2022)
- [Stable estimation of pulses of unknown shape from multiple snapshots via ESPRIT](https://mlsanity.com/paper/2308.16265/) (2023)

## Abstract (arXiv)

The problem of multi-snapshot spike deconvolution is studied, where the goal is to recover the locations of sparse impulses from their noisy convolution with a known point spread function (PSF) across multiple snapshots. A variable-projection formulation is adopted, in which the amplitudes are eliminated in closed form, thereby reducing the task to a nonconvex least-squares problem over the spike locations alone. This formulation is referred to as the variable-projection formulation of spike deconvolution (VarProSD). An explicit characterization of the basin of convexity of the VarProSD objective is provided in terms of key PSF properties, including its power spectral density and smoothness, revealing how sampling bandwidth and spike separation affect the local geometry. Within this basin, consistency of the estimator in the number of snapshots is established under stochastic noise, and a complementary, sharper error bound is derived under adversarial noise through the local Lipschitz property of the inverse map. Local convergence guarantees for gradient descent are further established when initialization is performed within the basin. A central role throughout the analysis is played by Beurling--Selberg extremal approximations, which enable sharp, PSF-agnostic bounds on the conditioning of the structured matrices arising in the optimization landscape. Numerical experiments are presented to corroborate the theoretical findings and demonstrate the effectiveness of modified ESPRIT initialization followed by gradient-based refinement.