2021/06/01 by Petr Mokrov, Alexander Korotin, Mokrov, Petr +9 · 10 citations
Computer Science · Mathematics · Medicine · #Advanced Neuroimaging Techniques and Applications #FOS: Computer and information sciences #Generative Adversarial Networks and Image Synthesis #Geometric Analysis and Curvature Flows #Machine Learning (cs.LG) #cs.LG
paper · pdf · doi:10.48550/arxiv.2106.00736
openalex publication_date 2021/06/01 · arxiv created 2021/10/25 · arxiv updated 2021/10/26 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Wasserstein gradient flows provide a powerful means of understanding and solving many diffusion equations. Specifically, Fokker-Planck equations, which model the diffusion of probability measures, can be understood as gradient descent over entropy functionals in Wasserstein space. This equivalence, introduced by Jordan, Kinderlehrer and Otto, inspired the so-called JKO scheme to approximate these diffusion processes via an implicit discretization of the gradient flow in Wasserstein space. Solving the optimization problem associated to each JKO step, however, presents serious computational challenges. We introduce a scalable method to approximate Wasserstein gradient flows, targeted to machine learning applications. Our approach relies on input-convex neural networks (ICNNs) to discretize the JKO steps, which can be optimized by stochastic gradient descent. Unlike previous work, our method does not require domain discretization or particle simulation. As a result, we can sample from the measure at each time step of the diffusion and compute its probability density. We demonstrate our algorithm's performance by computing diffusions following the Fokker-Planck equation and apply it to unnormalized density sampling as well as nonlinear filtering.