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Proximal methods for point source localisation

Tuomo Valkonen

TL;DR

This work extends proximal methods to spaces of Radon measures for point source localisation, addressing the limitations of Hilbert-space techniques. By introducing a particle-to-wave operator ${\mathscr{D}}$ to define a measure-distance and by exploiting convolution-based forward models, the authors derive forward-backward, inertial, and primal-dual proximal splitting algorithms with convergence guarantees. They show function-value descent and weak-$*$ convergence, plus ergodic convergence rates, under practical assumptions, and demonstrate numerical efficacy on 1D and 2D benchmark problems. The approach yields a grid-free or grid-friendly framework that can handle nonsmooth regularisation and nonsmooth data terms, offering a scalable alternative to Frank–Wolfe-type methods for measure-valued inverse problems.

Abstract

Point source localisation is generally modelled as a Lasso-type problem on measures. However, optimisation methods in non-Hilbert spaces, such as the space of Radon measures, are much less developed than in Hilbert spaces. Most numerical algorithms for point source localisation are based on the Frank-Wolfe conditional gradient method, for which ad hoc convergence theory is developed. We develop extensions of proximal-type methods to spaces of measures. This includes forward-backward splitting, its inertial version, and primal-dual proximal splitting. Their convergence proofs follow standard patterns. We demonstrate their numerical efficacy.

Proximal methods for point source localisation

TL;DR

This work extends proximal methods to spaces of Radon measures for point source localisation, addressing the limitations of Hilbert-space techniques. By introducing a particle-to-wave operator to define a measure-distance and by exploiting convolution-based forward models, the authors derive forward-backward, inertial, and primal-dual proximal splitting algorithms with convergence guarantees. They show function-value descent and weak- convergence, plus ergodic convergence rates, under practical assumptions, and demonstrate numerical efficacy on 1D and 2D benchmark problems. The approach yields a grid-free or grid-friendly framework that can handle nonsmooth regularisation and nonsmooth data terms, offering a scalable alternative to Frank–Wolfe-type methods for measure-valued inverse problems.

Abstract

Point source localisation is generally modelled as a Lasso-type problem on measures. However, optimisation methods in non-Hilbert spaces, such as the space of Radon measures, are much less developed than in Hilbert spaces. Most numerical algorithms for point source localisation are based on the Frank-Wolfe conditional gradient method, for which ad hoc convergence theory is developed. We develop extensions of proximal-type methods to spaces of measures. This includes forward-backward splitting, its inertial version, and primal-dual proximal splitting. Their convergence proofs follow standard patterns. We demonstrate their numerical efficacy.
Paper Structure (27 sections, 20 theorems, 121 equations, 5 figures, 4 algorithms)

This paper contains 27 sections, 20 theorems, 121 equations, 5 figures, 4 algorithms.

Table of Contents

  1. Introduction
  2. Distances of measures
  3. Wave-particle duality
  4. Particle-to-wave operators based on convolution
  5. Convolution kernels and sensor grids
  6. Sensor grids
  7. Examples
  8. Forward-backward splitting
  9. Optimality conditions and basic method
  10. Point insertion and reweighting
  11. Function value convergence
  12. Weak-$*$ convergence
  13. Inertial forward-backward
  14. Algorithm description
  15. Convergence
  16. ...and 12 more sections

Key Result

Lemma 2.1

Let $\rho \in C_0(\mathbb{R}^n)$ be symmetric. On a closed domain $\Omega \subset \mathbb{R}^n$, let ${\mathscr{D}} \mu := \rho * \mu$ for $\mu \in {\mathscr{M}}(\Omega)$. Then ${\mathscr{D}} \in \mathbb{L}({\mathscr{M}}(\Omega); C_0(\Omega))$ and is self-adjoint, i.e., satisfies eq:wave:self-adjoin

Figures (5)

  • Figure 1: Reconstructions and performance on 1D problem with cut Gaussian spread. Top: reconstruction and original data. The measurement data magnitude scale is on the right, spike magnitude on the left. Middle: Function value in terms of iteration count (left) and CPU time (right). The thin line indicates function value for $\mu$FB after postprocessing weight optimisation. Bottom: spike evolution, inner iteration count (left), and kernels (right). The thick lines indicate the spike count, and the thinner and dimmer lines the inner iteration count.
  • Figure 2: Reconstructions and performance on 1D problem with the “fast” spread. The plots are to be read the same way as \ref{['fig:1dproblem']}.
  • Figure 3: Reconstructions and performance on 2D problem with cut Gaussian spread. Top: reconstruction and original data. The area area of the top surface of the boxes is proportional the noise level of the underlying sensor, and their colour the sign of the noise. Middle: Function value in terms of iteration count and CPU time. The thin line indicates function value for $\mu$FB after postprocessing weight optimisation. Bottom: spike count evolution and kernels. The kernels have been shifted by $\pm 0.2$ in the $x$ and $y$ directions for visualisation-technical reasons.
  • Figure 4: Reconstructions and performance on 2D problem with the “fast” spread. The plots are to be read the same way as \ref{['fig:2dproblem']}.
  • Figure 5: Reconstructions and performance on 1D problem with salt and pepper noise, $\ell_1$ data term, and gaussian spread. The plots are to be read the same way as \ref{['fig:1dproblem']}.

Theorems & Definitions (57)

  • Lemma 2.1
  • proof
  • Example 2.2
  • Lemma 2.3
  • proof
  • Theorem 2.4
  • proof
  • Corollary 2.5
  • proof
  • Lemma 3.1
  • ...and 47 more