Optimal sampling density design for homotopic off-the-grid deconvolution

Joseph Gabet, Maxime Ferreira Da Costa, and Kiryung Lee

Abstract

Subsampled weighted Vandermonde matrices play a crucial role in spike deconvolution with a point spread function from random Fourier measurements. A prior theoretical analysis established bounds on the condition number and inspired an empirical study that successfully optimized those bounds to significantly improve conditioning. The optimized density multiplied by the power spectrum exhibits a distinctive flat structure across the sampling domain. This paper provides a theoretical foundation for this empirical observation. We prove that the continuous optimization problem admits an optimizer whose effective spectrum is a normalized indicator of finitely many intervals, which reduces to a centered ideal low pass filter when the spectrum is even and decreases away from the origin. This structural property naturally inspires a coarse to fine sequence of nested low pass variable projection objectives, with complexity bounded through covering number analysis. Experiments corroborate that this homotopic approach achieves superior separation performance over direct non homotopic refinement.

Cite

@misc{gabet:hal-05762978,
  author     = {Gabet, Joseph and Ferreira Da Costa, Maxime and Lee, Kiryung},
  title      = {Optimal sampling density design for homotopic off-the-grid deconvolution},
  year       = {2026},
  eprint     = {hal-05762978},
  eprinttype = {hal}
}