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

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

TL;DR

The paper tackles spike deconvolution from Fourier-domain measurements with a known PSF by formulating a non-convex, joint estimation problem for amplitudes $\bm{A}$ and spike locations $\btau$ via a non-linear least-squares loss $\mathcal{L}$. It introduces a two-stage framework that first initializes with a tailored ESPRIT method and then refines the parameters using a preconditioned gradient descent (PGD) with a full non-diagonal preconditioner, enabling supra-linear convergence. The authors prove local convergence of PGD under a separation condition $\Delta > (2/3)\rho_{g'}$, derive stability bounds for ESPRIT via a Davis--Kahan analysis, and show that, in 1D with multiple snapshots and sufficiently high SNR, ESPRIT initialization yields global convergence when followed by PGD. Numerical results corroborate the theory, showing improved spike-location accuracy for narrow PSFs and higher SNR, and establish the practical viability and robustness of the approach in Fourier-domain spike deconvolution.

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.

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

TL;DR

The paper tackles spike deconvolution from Fourier-domain measurements with a known PSF by formulating a non-convex, joint estimation problem for amplitudes and spike locations via a non-linear least-squares loss . It introduces a two-stage framework that first initializes with a tailored ESPRIT method and then refines the parameters using a preconditioned gradient descent (PGD) with a full non-diagonal preconditioner, enabling supra-linear convergence. The authors prove local convergence of PGD under a separation condition , derive stability bounds for ESPRIT via a Davis--Kahan analysis, and show that, in 1D with multiple snapshots and sufficiently high SNR, ESPRIT initialization yields global convergence when followed by PGD. Numerical results corroborate the theory, showing improved spike-location accuracy for narrow PSFs and higher SNR, and establish the practical viability and robustness of the approach in Fourier-domain spike deconvolution.

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.

Paper Structure

This paper contains 9 sections, 3 theorems, 23 equations, 2 figures, 1 algorithm.

Table of Contents

  1. Introduction
  2. Contributions, Prior Art, and Organization of the Paper
  3. Notation and Definitions
  4. Problem Formulation
  5. Problem conditioning and critical metrics
  6. Local Convergence of Preconditioned Gradient Descent
  7. ESPRIT Initialization Using Known PSF
  8. Global Convergence of PGD Initialized with ESPRIT and Numerical Results
  9. Conclusion

Key Result

Theorem 1

Suppose $\Delta > \frac{2}{3}\rho_{g'}$ and let the quantities If $4(\alpha+1)\beta \left\vert u_{\min}^\star \right\vert^{-1} \left\Vert \bm{Z} \right\Vert_F \leq 1$ and the initialization $\bm{\theta}_0$ has a weighted error $\eta_0$ satisfying then the error sequence ${\{\eta_k\}}_k$ satisfies Furthermore, ${\{\eta_k\}}_k$ converges super-linearly into $[0,\gamma_\infty]$.

Figures (2)

  • Figure 1: Illustration of the convergence of PGD. The weighted error sequence satisfies $\eta_{k+1} \leq f_1(\eta_k)$ where $f_1(\eta) = \frac{\alpha \eta^2 + \beta |u^\star_{\min}|^{-1}\|\bZ\|_F}{1-\eta}$ (solid;green). When the sequence $(\eta_k)_k$ starts between the two fixed points of $f_1(\eta)$ and $f_2(\eta) = \eta$ (dashed;red), its upper bound $(\gamma_k)_k$ converges to the left fixed point $\gamma_\infty$ super linearly.
  • Figure 2: Performance of ESPRIT and ESPRIT plus PGD.

Theorems & Definitions (3)

  • Theorem 1: Local convergence of PGD
  • Theorem 2
  • Lemma 1