337. Application of the Opial property in Wasserstein spaces to inexact JKO schemes
Invited abstract in session FD-7: Optimal Transport for Machine Learning and Inverse Problems, stream Optimization for Inverse Problems and Machine Learning.
Friday, 14:10 - 15:50Room: M:I
Authors (first author is the speaker)
| 1. | Emanuele Naldi
|
| Mathematics, Università di Genova |
Abstract
The Opial property is a metric characterization of weak convergence for a suitable class of Banach spaces. It plays an important role in the study of weak convergence of iterates of mappings and of the asymptotic behavior of nets satisfying some metric properties. Since it involves only metric quantities, it is possible to define this property also in metric spaces provided with a suitable notion of weak convergence. This is the case for the space of probability measures endowed with the Kantorovich-Rubinstein-Wasserstein metric deriving by optimal transport. In this talk, we present the Opial property in the Wasserstein space of Borel probability measures with finite quadratic moment on a separable Hilbert space. We present applications of this property to convergence of sequences generated by the JKO scheme (the proximal point algorithm in Wasserstein spaces) when the functional is lower semicontinuous and convex along generalized geodesics. In practice, the computation of a proximal step in Wasserstein spaces can be challenging and it is often carried out only approximately. For this reason, we discuss various type of inexactness that can be introduced in the method, showing convergence and providing rates.
Keywords
- Convex and non-smooth optimization
Status: accepted
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