Point source localisation with unbalanced optimal transport

TL;DR

This work develops forward-backward type optimization in spaces of Radon measures for point-source localisation by replacing the Hilbert-space proximal penalty with an unbalanced optimal transport distance. It avoids computing full OT plans by using transport three-plans and transport subdifferentials, coupling measure updates with a particle-to-wave marginal operator and a weak-$*$ metrisation to enable efficient iterations. Theoretical results establish existence, weak-$*$ convergence, and sublinear functional value convergence (ergodic $O(1/N)$) under reasonable assumptions, while extensions to product spaces and primal-dual schemes broaden applicability. Numerical experiments show that sliding proximal methods based on the $\mathscr{D}$-norm outperform Radon-norm proximal variants and conventional methods in 1D/2D localization tasks, validating the practical impact of transport-based regularisation for sparse measure recovery.

Abstract

Replacing the quadratic proximal penalty familiar from Hilbert spaces by an unbalanced optimal transport distance, we develop forward-backward type optimisation methods in spaces of Radon measures. We avoid the actual computation of the optimal transport distances through the use of transport three-plans and the rough concept of transport subdifferentials. The resulting algorithm has a step similar to the sliding heuristics previously introduced for conditional gradient methods, however, now non-heuristically derived from the geometry of the space. We demonstrate the improved numerical performance of the approach.

Figures (4)

• Figure 1: Reconstructions and performance on 1D problem with “fast” 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). 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 2D problem with “fast” spread. Top: reconstruction and original data. The 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. Bottom: spike and inner iteration count evolution, and kernels.
• Figure 3: Reconstructions and performance on 1D problem with “fast” spread and TV-regularised bias. 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). 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 4: Reconstructions and performance on 2D problem with “fast” spread and TV-regularised bias. Top: reconstruction and original data. The upper layer displays the data, noise level, and spike reconstructions, while the lower layer displays the bias and its reconstruction by sPDPS. The area of the top surface of the boxes that indicate the noise level 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. Bottom: spike and inner iteration count evolution, and kernels.

Theorems & Definitions (82)

• Lemma 2.1
• proof
• Lemma 2.2
• proof
• Lemma 2.3
• proof
• Corollary 2.4
• proof
• Corollary 2.5
• proof