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Projected Block Coordinate Descent for sparse spike estimation

Pierre-Jean Bénard, Yann Traonmilin, Jean François Aujol

TL;DR

This work tackles the problem of recovering off-the-grid sparse spikes from linear measurements and builds on the OP-COMP+PGD framework. It introduces a Projected Block Coordinate Descent (BCD) that partitions the estimation problem into spike-based blocks and updates only the non-converged blocks using a Gauss-Southwell-inspired rule, with a FISTA Restart for the inner descent. The authors provide a qualitative block-selection argument based on dipole decomposition and mutual coherence, and demonstrate a practical speedup of up to 35% in MA-TIRF microscopy experiments without sacrificing reconstruction quality. The approach offers a scalable acceleration for sparse spike recovery with potential parallelization and broader applicability to similar non-convex, over-parametrized recovery problems.

Abstract

We consider the problem of recovering off-the-grid spikes from linear measurements. The state of the art Over-Parametrized Continuous Orthogonal Matching Pursuit (OP-COMP) with Projected Gradient Descent (PGD) successfully recovers those signals. In most cases, the main computational cost lies in a unique global descent on all parameters (positions and amplitudes). In this paper, we propose to improve this algorithm by accelerating this descent step. We introduce a new algorithm, based on Block Coordinate Descent, that takes advantages of the sparse structure of the problem. Based on qualitative theoretical results, this algorithm shows improvement in calculation times in realistic synthetic microscopy experiments.

Projected Block Coordinate Descent for sparse spike estimation

TL;DR

This work tackles the problem of recovering off-the-grid sparse spikes from linear measurements and builds on the OP-COMP+PGD framework. It introduces a Projected Block Coordinate Descent (BCD) that partitions the estimation problem into spike-based blocks and updates only the non-converged blocks using a Gauss-Southwell-inspired rule, with a FISTA Restart for the inner descent. The authors provide a qualitative block-selection argument based on dipole decomposition and mutual coherence, and demonstrate a practical speedup of up to 35% in MA-TIRF microscopy experiments without sacrificing reconstruction quality. The approach offers a scalable acceleration for sparse spike recovery with potential parallelization and broader applicability to similar non-convex, over-parametrized recovery problems.

Abstract

We consider the problem of recovering off-the-grid spikes from linear measurements. The state of the art Over-Parametrized Continuous Orthogonal Matching Pursuit (OP-COMP) with Projected Gradient Descent (PGD) successfully recovers those signals. In most cases, the main computational cost lies in a unique global descent on all parameters (positions and amplitudes). In this paper, we propose to improve this algorithm by accelerating this descent step. We introduce a new algorithm, based on Block Coordinate Descent, that takes advantages of the sparse structure of the problem. Based on qualitative theoretical results, this algorithm shows improvement in calculation times in realistic synthetic microscopy experiments.
Paper Structure (7 sections, 1 theorem, 16 equations, 3 figures, 1 algorithm)

This paper contains 7 sections, 1 theorem, 16 equations, 3 figures, 1 algorithm.

Table of Contents

  1. Introduction
  2. The Projected Block Coordinate Descent algorithm
  3. A result to guide block selection
  4. Description of the algorithm
  5. Experiments
  6. Discussion/Conclusion
  7. Acknowledgments

Key Result

Lemma 1

Let $x_{0} = \sum_{i = 1}^{K} a_{i} \delta_{t_{i}} \in \Sigma_{K, \epsilon}$ and $x = \phi(\theta) = \sum_{i = 1}^{K} b_{i} \delta_{s_{i}}$ with $\theta = (b_{1}, \dots, b_{K}, s_{1}, \dots, s_{K})$ such that for all $i \in \{ 1, \dots, K \}$, $\| t_{i} - s_{i} \|_{2} < \frac{\epsilon}{3}$. Let $A$ Then we obtain Moreover, let $\partial_{i, r} g(\theta)$ be the partial derivative of $g(\theta)$

Figures (3)

  • Figure 1: (a) Ground truth and a plane of its observation through MA-TIRF, (b) Ground truth and the initialized signal by OP-COMP.
  • Figure 2: (a) Ground truth and the estimated signal by PGD with MA-TIRF, (b) Ground truth and the estimated signal by BCD with MA-TIRF, (c) Comparison norms of residues by PGD and BCD in function of time.
  • Figure 3: (a) Fraction of spikes whose gradients are computed during BCD, (b) Time elapsed for PGD and BCD in function of the number of iterations

Theorems & Definitions (4)

  • Definition 1: ($\epsilon$-)Dipole
  • Definition 2: ($\epsilon$)-Separation of dipoles
  • Lemma 1
  • proof