mlsanityData to 7 Oct 2026

Non-geodesically-convex optimization in the Wasserstein space

arXiv 2406.00502 ↗First posted 1 Jun 2024math.OC cs.LGcited by 5

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Abstract (arXiv)

We study a class of optimization problems in the Wasserstein space (the space of probability measures) where the objective function is nonconvex along generalized geodesics. Specifically, the objective exhibits some difference-of-convex structure along these geodesics. The setting also encompasses sampling problems where the logarithm of the target distribution is difference-of-convex. We derive multiple convergence insights for a novel semi Forward-Backward Euler scheme under several nonconvex (and possibly nonsmooth) regimes. Notably, the semi Forward-Backward Euler is just a slight modification of the Forward-Backward Euler whose convergence is -- to our knowledge -- still unknown in our very general non-geodesically-convex setting.

Key facts. Cited by 5 papers in the map (data through 7 Oct 2026). Builds on 31 of its 78 references that are in the map. First posted on arXiv 1 Jun 2024.

Builds on

31 of 78 references.

References · 31

  1. 2016Linear Convergence of Gradient and Proximal-Gradient Methods Under the Polyak-Lojasiewicz Conditioncited by 526
  2. 2018On the Global Convergence of Gradient Descent for Over-parameterized Models using Optimal Transportcited by 422
  3. 2016Input Convex Neural Networkscited by 407
  4. 2019Rapid Convergence of the Unadjusted Langevin Algorithm: Isoperimetry Sufficescited by 224
  5. 2014Theoretical guarantees for approximate sampling from smooth and log-concave densitiescited by 206
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  7. 2019Mean-field theory of two-layers neural networks: dimension-free bounds and kernel limitcited by 187
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  11. 2018Sampling as optimization in the space of measures: The Langevin dynamics as a composite optimization problemcited by 123
  12. 2019Maximum Mean Discrepancy Gradient Flowcited by 123
19 more
  1. 2019Wasserstein-2 Generative Networkscited by 96
  2. 2022Variational inference via Wasserstein gradient flowscited by 84
  3. 2016Efficient Bayesian computation by proximal Markov chain Monte Carlo: when Langevin meets Moreaucited by 83
  4. 2021Large-Scale Wasserstein Gradient Flowscited by 76
  5. 2022Towards a Theory of Non-Log-Concave Sampling: First-Order Stationarity Guarantees for Langevin Monte Carlocited by 63
  6. 2021Optimizing Functionals on the Space of Probabilities with Input Convex Neural Networkscited by 53
  7. 2021Variational Wasserstein gradient flowcited by 53
  8. 2018On the Fenchel Duality between Strong Convexity and Lipschitz Continuous Gradientcited by 42
  9. 2023Forward-backward Gaussian variational inference via JKO in the Bures-Wasserstein Spacecited by 36
  10. 2022First-order Conditions for Optimization in the Wasserstein Spacecited by 19
  11. 2019Accelerated Flow for Probability Distributionscited by 15
  12. 2015Computing B-Stationary Points of Nonsmooth DC Programscited by 12
  13. 2018Composite Difference-Max Programs for Modern Statistical Estimation Problemscited by 10
  14. 2014DC approximation approaches for sparse optimizationcited by 8
  15. 2018A Pontryagin Maximum Principle in Wasserstein Spaces for Constrained Optimal Control Problemscited by 7
  16. 2017On the Pervasiveness of Difference-Convexity in Optimization and Statisticscited by 4
  17. 2019Low-rank matrix recovery with Ky Fan 2-k-normcited by 3
  18. 2022A Boosted-DCA with Power-Sum-DC Decomposition for Linearly Constrained Polynomial Programscited by 3
  19. 2022Distance-to-Set Priors and Constrained Bayesian Inferencecited by 1

Built on by

Cited by year (2026 to 7 Oct)

Citing papers per year: 2024: 2; 2025: 1; 2026: 2.

5 papers.

  1. 2024Linear convergence of proximal descent schemes on the Wasserstein spacecited by 3
  2. 2024Stochastic variance-reduced Gaussian variational inference on the Bures-Wasserstein manifoldcited by 2
  3. 2025Convergence Analysis of the Wasserstein Proximal Algorithm beyond Geodesic Convexitycited by 1
  4. 2026DC-LA: Difference-of-Convex Langevin Algorithmcited by 1
  5. 2026Difference of Convex Programming in the Wasserstein Space with Applications to MMD Optimizationcited by 0