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Proximal methods for point source localisation

arXiv 2212.02991 ↗First posted 6 Dec 2022math.OC cs.CVcited by 0

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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.

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References · 14

  1. 2018On the Global Convergence of Gradient Descent for Over-parameterized Models using Optimal Transportcited by 426
  2. 2019Sparse Optimization on Measures with Over-parameterized Gradient Descentcited by 50
  3. 2021Convergence Rates of Gradient Methods for Convex Optimization in the Space of Measurescited by 20
  4. 2018The Sliding Frank-Wolfe Algorithm and its Application to Super-Resolution Microscopycited by 16
  5. 2020Introduction to Nonsmooth Analysis and Optimizationcited by 4
  6. 2020A generalized conditional gradient method for dynamic inverse problems with optimal transport regularizationcited by 4
  7. 2017Testing and non-linear preconditioning of the proximal point methodcited by 3
  8. 2019Linear convergence of accelerated conditional gradient algorithms in spaces of measurescited by 3
  9. 2019On the linear convergence rates of exchange and continuous methods for total variation minimizationcited by 3
  10. 2019First-order primal-dual methods for nonsmooth nonconvex optimisationcited by 2
  11. 2015An extension of the projected gradient method to a Banach space setting with application in structural topology optimizationcited by 1
  12. 2017Preconditioned proximal point methods and notions of partial subregularitycited by 1
2 more
  1. 2018Inertial, corrected, primal-dual proximal splittingcited by 1
  2. 2020Regularisation, optimisation, subregularitycited by 1

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