# Proximal methods for point source localisation

arXiv: https://arxiv.org/abs/2212.02991 · HTML: https://mlsanity.com/paper/2212.02991/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 14 of its 43 references that are in the map.
- First posted on arXiv 6 Dec 2022.

## Builds on (most cited first, 14 of 14)

- [On the Global Convergence of Gradient Descent for Over-parameterized Models using Optimal Transport](https://mlsanity.com/paper/1805.09545/) (2018)
- [Sparse Optimization on Measures with Over-parameterized Gradient Descent](https://mlsanity.com/paper/1907.10300/) (2019)
- [Convergence Rates of Gradient Methods for Convex Optimization in the Space of Measures](https://mlsanity.com/paper/2105.08368/) (2021)
- [The Sliding Frank-Wolfe Algorithm and its Application to Super-Resolution Microscopy](https://mlsanity.com/paper/1811.06416/) (2018)
- [Introduction to Nonsmooth Analysis and Optimization](https://mlsanity.com/paper/2001.00216/) (2020)
- [A generalized conditional gradient method for dynamic inverse problems with optimal transport regularization](https://mlsanity.com/paper/2012.11706/) (2020)
- [Testing and non-linear preconditioning of the proximal point method](https://mlsanity.com/paper/1703.05705/) (2017)
- [Linear convergence of accelerated conditional gradient algorithms in spaces of measures](https://mlsanity.com/paper/1904.09218/) (2019)
- [On the linear convergence rates of exchange and continuous methods for total variation minimization](https://mlsanity.com/paper/1906.09919/) (2019)
- [First-order primal-dual methods for nonsmooth nonconvex optimisation](https://mlsanity.com/paper/1910.00115/) (2019)
- [An extension of the projected gradient method to a Banach space setting with application in structural topology optimization](https://mlsanity.com/paper/1503.03783/) (2015)
- [Preconditioned proximal point methods and notions of partial subregularity](https://mlsanity.com/paper/1711.05123/) (2017)
- [Inertial, corrected, primal-dual proximal splitting](https://mlsanity.com/paper/1804.08736/) (2018)
- [Regularisation, optimisation, subregularity](https://mlsanity.com/paper/2011.07575/) (2020)

## Abstract (arXiv)

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.