Global Convergence of ESPRIT with Preconditioned First-Order Methods for Spike Deconvolution

Joseph Gabet, Meghna Kalra, Maxime Ferreira Da Costa, and Kiryung Lee

European Signal Processing Conference (EUSIPCO 2026), to appear.

Abstract

Spike deconvolution is the problem of recovering point sources from their convolution with a known point spread function, playing a fundamental role in many sensing and imaging applications. This paper proposes a novel approach combining ESPRIT with Preconditioned Gradient Descent (PGD) to estimate the amplitudes and locations of the point sources by a non-linear least squares. The preconditioning matrices are adaptively designed to account for variations in the learning process, ensuring a proven super-linear convergence rate. We provide local convergence guarantees for PGD and performance analysis of ESPRIT reconstruction, leading to global convergence guarantees for our method in one-dimensional settings with multiple snapshots, demonstrating its robustness and effectiveness. Numerical simulations corroborate the performance of the proposed approach for spike deconvolution.

Cite

@inproceedings{gabet2025GlobalConvergencea,
  author     = {Gabet, Joseph and Kalra, Meghna and Ferreira Da Costa, Maxime and Lee, Kiryung},
  title      = {Global Convergence of {{ESPRIT}} with Preconditioned First-Order Methods for Spike Deconvolution},
  booktitle  = {2026 European Signal Processing Conference (EUSIPCO)},
  year       = {2026},
  doi        = {10.48550/arXiv.2502.08035},
  eprint     = {2502.08035},
  eprinttype = {arXiv}
}