Variational and information flows in machine learning and optimal transport / Wuchen Li, Bernhard Schmitzer, Gabriele Steidl, François-Xavier Vialard, Christian Wald.

Format:

Book

Author/Creator:

Li, Wuchen (Mathematician), author.

Schmitzer, Bernhard (Computer scientist), author.

Steidl, Gabriele, author.

Vialard, François-Xavier, author.

Wald, Christian, author.

Series:

Oberwolfach seminars ; v. 56.

Oberwolfach seminars, 1661-237x ; volume 56

Language:

English

Subjects (All):

Stochastic processes.

Computational intelligence.

Machine learning--Mathematics.

Machine learning.

Variational inequalities (Mathematics).

Physical Description:

xiv, 254 pages : illustrations (black and white, and colour) ; 24 cm.

Place of Publication:

Cham, Switzerland : Birkhäuser, Springer Nature, [2025]

Summary:

This book is based on lectures given at the Mathematisches Forschungsinstitut Oberwolfach on "Computational Variational Flows in Machine Learning and Optimal Transport". Variational and stochastic flows on measure spaces are ubiquitous in machine learning and generative modeling. Optimal transport and diffeomorphic flows provide powerful frameworks to analyze such trajectories of distributions with elegant notions from differential geometry, such as geodesics, gradient and Hamiltonian flows. Recently, mean field control and mean field games offered a general optimal control variational view on learning problems. The four independent chapters in this book address the question of how the presented tools lead us to better understanding and further development of machine learning and generative models.

Contents:

A dynamic perspective of optimal transport / Bernhard Schmitzer

A geometric perspective on diffeomorphic and optimal transport flows and their applications / François-Xavier Vialard

3. Generalized Wasserstein dynamics in mathematical data sciences / Wuchen Li

Flow matching: Markov kernels, stochastic processes and transport plans / Christian Wald and Gabriele Steidl

Appendix A. Proof of theorm 3.1

Appendix B. Measurability of v[subscript]t in Lemma 5.7

Appendix C. Remakrs on normalizing flows

References.

ISBN:

9783031927300

3031927303

OCLC:

1528631080

Publisher Number:

CIPO000253498