{Euclidean, metric, and Wasserstein} gradient flows: an overview

By Filippo Santambrogio. 2016

338 citations recorded by the source corpus.

This is an expository paper on the theory of gradient flows, and in particular of those PDEs which can be interpreted as gradient flows for the Wasserstein metric on the space of probability measures (a distance induced by optimal transport). The starting point is the Euclidean theory, and then its generalization to metric spaces, according to the work of Ambrosio, Gigli and Savaré. Then comes an independent exposition of the Wasserstein theory, with a short introduction to the optimal transport tools that are needed and to the notion of geodesic convexity, followed by a precise description of the Jordan–Kinderlehrer–Otto scheme and a sketch of proof to obtain its convergence in the easiest cases. A discussion of which equations are gradient flows PDEs and of numerical methods based on these ideas is also provided. The paper ends with a new, theoretical, development, due to Ambrosio, Gigli, Savaré, Kuwada and Ohta: the study of the heat flow in metric measure spaces.

Subfields: Applied Mathematics, Astronomy and Astrophysics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/cba209da222a2ef655abfedd7a45252fe876fb2e

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Applied Mathematics

1. G. Peyr'e
2. Marco Cuturi
3. Pratik Chaudhari
4. Chang Liu
5. Stefano Soatto
6. G. Em Karniadakis
7. Jingwei Zhuo
8. M. L. Ang

In Astronomy and Astrophysics

1. Pratik Chaudhari
2. G. Peyr'e
3. Marco Cuturi
4. Stefano Soatto
5. G. Em Karniadakis
6. Liu Yang

In Geometry and Topology

1. Pratik Chaudhari
2. G. Peyr'e
3. Chang Liu
4. Marco Cuturi
5. Stefano Soatto
6. G. Em Karniadakis
7. Jingwei Zhuo
8. Anima Anandkumar

Citations in the corpus

• Understanding MCMC Dynamics as Flows on the Wasserstein Space (2019, background): “According to existing knowledge (e.g. Villani (2008), Ex 15.10; Ambrosio et al. (2008), Lem 10.4.1; Santambrogio (2017), Eq 4.10), its gradient at q0 is given by( gradF ) (q0) = grad ( δF δq ∣∣∣∣ q=q0 ) , where δFδq is the first functional variation of F , which is log(q0/p) at q = q0.”
• Meta-Learning with Neural Tangent Kernels (2021, neutral): “Furthermore, for arbitrary T > 0 we have: , which is the resulting function whose parameters evolve according to the gradient flow d θ We actually have the following (Santambrogio, 2016): For simplicity and clearness, we use (cid:52) to denote the change of any vectors and matrices.”
• Stochastic Gradient Descent Performs Variational Inference, Converges to Limit Cycles for Deep Networks (2017, methodology): “The proof follows from Theorem 5 and Lemma 6, see [9] for a rigorous treatment of Wasserstein metrics.”
• Computational Optimal Transport (2018, result): “We also refer to the recent survey paper [Santambrogio, 2017].”
• Refining Deep Generative Models via Discriminator Gradient Flow (2020, background): “The following gives a brief introduction to gradient flows; we refer readers to the excellent overview by Santambrogio (2017) for a more thorough introduction.”
• Controllable and Compositional Generation with Latent-Space Energy-Based Models (2021, background): “Similar to [45, 3] that build connections between SDE/ODE and gradient flows, our ODE sampler can be seen as a gradient flow that refines a latent z from a random noise to a z vector conditioned on an attribute vector.”
• Potential Flow Generator With L2 Optimal Transport Regularity for Generative Models (2019, background): “For example, it is closely related to gradient flow [Ambrosio et al., 2008], Fokker-Planck equations [Santambrogio, 2017], porous medium flow [Otto, 1997], etc. 3 Potential Flow Generator”