# Weak topology and Opial property in Wasserstein spaces, with applications to Gradient Flows and Proximal Point Algorithms of geodesically convex functionals

arXiv: https://arxiv.org/abs/2104.06121 · HTML: https://mlsanity.com/paper/2104.06121/

## Key facts

- Cited by 4 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 13 Apr 2021.

## Built on by (most cited first, 4 of 4)

- [Forward-backward Gaussian variational inference via JKO in the Bures-Wasserstein Space](https://mlsanity.com/paper/2304.05398/) (2023)
- [Wasserstein Gradient Flows of MMD Functionals with Distance Kernel and Cauchy Problems on Quantile Functions](https://mlsanity.com/paper/2408.07498/) (2024)
- [A Lipschitz spaces view of infinitely wide shallow neural networks](https://mlsanity.com/paper/2410.14591/) (2024)
- [Convergence Analysis of the Wasserstein Proximal Algorithm beyond Geodesic Convexity](https://mlsanity.com/paper/2501.14993/) (2025)

## Abstract (arXiv)

In this paper we discuss how to define an appropriate notion of weak topology in the Wasserstein space (P2(H),W2) of Borel probability measures with finite quadratic moment on a separable Hilbert space H. We will show that such a topology inherits many features of the usual weak topology in Hilbert spaces, in particular the weak closedness of geodesically convex closed sets and the Opial property characterizing weakly convergent sequences. We apply this notion to the approximation of fixed points for a non-expansive map in a weakly closed subset of P2(H) and of minimizers of a lower semicontinuous and geodesically convex functional φ:P2(H)→(-∞,+∞] attaining its minimum. In particular, we will show that every solution to the Wasserstein gradient flow of φ weakly converge to a minimizer of φ as the time goes to +∞. Similarly, if φ is also convex along generalized geodesics, every sequence generated by the proximal point algorithm converges to a minimizer of φ with respect to the weak topology of P2(H).