Gradient Flows: In Metric Spaces and in the Space of Probability Measures

By Luigi Ambrosio, Nicola Gigli, Giuseppe Savar'e. 2005

4,314 citations recorded by the source corpus.

How researchers see this paper

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

Citations in the corpus

Model Fusion via Optimal Transport (2019, background): “Note that, when the underlying ground cost used is the squared Euclidean distance, this barycentric mapping is known to be optimal [33].”
Improving Task-free Continual Learning by Distributionally Robust Memory Evolution (2022, methodology): “(Ambrosio et al., 2008).”
On the Global Convergence of Gradient Descent for Over-parameterized Models using Optimal Transport (2018, methodology): “The term involving V , well-studied in the theory of Wasserstein gradient ﬂows (see [3, Prop.”
Computational Optimal Transport (2018, background): “27) is a special case of a more general formalism to define gradient flows over metric spaces (X , d), where d is a distance, as detailed in [Ambrosio et al., 2006].”
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.”
Zero-sum stochastic differential games of generalized McKean–Vlasov type (2018, unknown): “[2], [16]), as recently shown in [17].”
BV Estimates in Optimal Transportation and Applications (2015, background): “This is equivalent (see, for instance, [2] or [25]) to (5.”
Riemannian metrics for neural networks I: feedforward networks (2013, background): “…yielding the smallest values of f , penalized by the distance from the current point.7 This makes it clear how the choice of the scalar product will influence 7 This can be used to define or study the direction of the gradient in more general metric spaces (e.g. Ambrosio et al., 2005, Chapter 2).”
Finsler interpolation inequalities (2009, methodology): “We also refer to [AGS], [RR], [Vi1] and [Vi2] for background information and further developments.”
Heat flow on Finsler manifolds (2008, methodology): “Since E0 is a convex function on the Hilbert space L2(M), we can apply Crandall and Liggett’s classical technique [CL] (see also [Ma], [AGS] for generalizations to curved spaces).”
A MACROSCOPIC CROWD MOTION MODEL OF GRADIENT FLOW TYPE (2010, methodology): “All the theory of Gradient Flow in Wasserstein Spaces is treated in the reference book by Ambrosio-Gigli-Savaré[2] and one of the key assumptions is the λ–convexity of the functional, which ensures better estimates.”
Nonsmooth analysis of doubly nonlinear evolution equations (2011, background): “, [17, 12, 6, 1, 47, 48, 41, 2, 44]), and in general doubly nonlinear equations [16, 15, 52, 51, 36], we approximate (2.”