12,649 Hits in 7.8 sec

Riemannian Convex Potential Maps [article]

Samuel Cohen, Brandon Amos, Yaron Lipman
2021 arXiv   pre-print
The budding approaches in this space are limited by representational and computational tradeoffs. We propose and study a class of flows that uses convex potentials from Riemannian optimal transport.  ...  We demonstrate that these flows can model standard distributions on spheres, and tori, on synthetic and geological data.  ...  Acknowledgments We thank Ricky Chen, Laurent Dinh, Maximilian Nickel and Marc Deisenroth for insightful discussions and acknowledge the Python community (Van Rossum & Drake Jr, 1995; Oliphant, 2007)  ... 
arXiv:2106.10272v1 fatcat:lr7agboeanh57luqumwqqiifoq

Large-Scale Wasserstein Gradient Flows [article]

Petr Mokrov, Alexander Korotin, Lingxiao Li, Aude Genevay, Justin Solomon, Evgeny Burnaev
2021 arXiv   pre-print
Our approach relies on input-convex neural networks (ICNNs) to discretize the JKO steps, which can be optimized by stochastic gradient descent.  ...  Wasserstein gradient flows provide a powerful means of understanding and solving many diffusion equations.  ...  JKO Reformulation via Optimal Push-forwards Maps Our key idea is to replace the optimization (6) over probability measures by an optimization over convex functions, an idea inspired by [10] .  ... 
arXiv:2106.00736v2 fatcat:dimf5e6zunbspe2uondxo532mi

Optimal transport by omni-potential flow and cosmological reconstruction

Uriel Frisch, Olga Podvigina, Barbara Villone, Vladislav Zheligovsky
2012 Journal of Mathematical Physics  
Mappings with a convex potential are known to be associated with the quadratic-cost optimal transport problem.  ...  a convex potential, a property we call omni-potentiality.  ...  Mappings with a convex potential are known to be associated with the quadratic-cost optimal transport problem.  ... 
doi:10.1063/1.3691203 fatcat:dgtwbqkvb5c5lkpbnulqkr343q

Mass transport generated by a flow of Gauss maps

Vladimir I. Bogachev, Alexander V. Kolesnikov
2009 Journal of Functional Analysis  
Moreover, T is invertible and essentially unique. Our proof employs the optimal transportation techniques.  ...  We prove that there exists a mapping T such that ν = µ • T −1 and T = ϕ · n, where ϕ : A → [0, r] is a continuous potential with convex sub-level sets and n is the Gauss map of the corresponding level  ...  Our work is motivated by two intensively developing areas: optimal transportation and curvature flows, and establishes an interesting link between these areas.  ... 
doi:10.1016/j.jfa.2008.05.006 fatcat:rnlaopvvz5d2vehncdp5ireaiq

Lagrangian schemes for Wasserstein gradient flows [article]

Jose A. Carrillo and Daniel Matthes and Marie-Therese Wolfram
2020 arXiv   pre-print
Keller-Segel model and of the fourth order thin film equation.  ...  This paper reviews different numerical methods for specific examples of Wasserstein gradient flows: we focus on nonlinear Fokker-Planck equations,but also discuss discretizations of the parabolic-elliptic  ...  In fact, the aforementioned convex function ϕ which extends the values of ϕ i at the x i is a Kantorovich potential for this transport.  ... 
arXiv:2003.03803v1 fatcat:i7cooreodjhudcevaifii23hka

The Bayesian update: variational formulations and gradient flows [article]

Nicolas Garcia Trillos, Daniel Sanz-Alonso
2018 arXiv   pre-print
Each gradient flow naturally suggests a nonlinear diffusion with the posterior as invariant distribution.  ...  By construction, the diffusions are guaranteed to satisfy a certain optimality condition, and rates of convergence are given by the convexity of the functionals.  ...  Data" that took place at Carnegie Mellon University in March 2017.  ... 
arXiv:1705.07382v2 fatcat:nnc4yuzivnfkzhpkzculhgobey

Propagation of chaos, Wasserstein gradient flows and toric Kahler-Einstein metrics [article]

Robert J. Berman, Magnus Onnheim
2016 arXiv   pre-print
This drift-diffusion equation is the gradient flow on the Wasserstein space of probability measures of the K-energy functional in Kahler geometry and it can be seen as a fully non-linear version of various  ...  (possibly singular) quasi-convex N-particle interaction energy.  ...  By the results in [9] the assumptions in Theorem 1.1 hold with E(µ) := −C(µ), where C(µ) is the Monge-Kantorovich optimal cost for transporting µ to the uniform probability measure ν P on the convex  ... 
arXiv:1501.07820v3 fatcat:2oymvyj5ifb47czrkiuajzd54e

On the velocities of flows consisting of cyclically monotone maps

A. Tudorascu
2010 Indiana University Mathematics Journal  
to the corresponding family of optimal maps pushing forward a given reference measure to each measure on the curve.  ...  Motivated by work on one-dimensional Euler-Poisson systems, Gangbo et al. proved a surprisingly general flow-map formula which unequivocally links an absolutely continuous curve in the Wasserstein space  ...  Nguyen and W. Gangbo for their valuable comments and suggestions.  ... 
doi:10.1512/iumj.2010.59.3955 fatcat:j6l6mi44m5bkphqyxzdkzccp7m

On the signed porous medium flow

Edoardo Mainini
2012 Networks and Heterogeneous Media  
We prove that the signed porous medium equation can be regarded as limit of an optimal transport variational scheme, therefore extending the classical result for positive solutions of [13] and showing  ...  that an optimal transport approach is suited even for treating signed densities.  ...  on the space as the standard optimal transport problem for probabilities.  ... 
doi:10.3934/nhm.2012.7.525 fatcat:7ajwms3s3jh2fabntkq2it7r2u

KALE Flow: A Relaxed KL Gradient Flow for Probabilities with Disjoint Support [article]

Pierre Glaser, Michael Arbel, Arthur Gretton
2021 arXiv   pre-print
Like the MMD and other Integral Probability Metrics, the KALE remains well defined for mutually singular distributions.  ...  We study the gradient flow for a relaxed approximation to the Kullback-Leibler (KL) divergence between a moving source and a fixed target distribution.  ...  Introduction We consider the problem of transporting probability mass from a source distribution P to a target distribution Q using a Wasserstein gradient flow in probability space.  ... 
arXiv:2106.08929v2 fatcat:xbmhbpxrafe7lfizpqy5zmcvea

Policy Optimization as Wasserstein Gradient Flows [article]

Ruiyi Zhang, Changyou Chen, Chunyuan Li, Lawrence Carin
2018 arXiv   pre-print
On the probability-measure space, under specified circumstances, policy optimization becomes a convex problem in terms of distribution optimization.  ...  We place policy optimization into the space of probability measures, and interpret it as Wasserstein gradient flows.  ...  Acknowledgements We acknowledge Tuomas Haarnoja et al. for making their code public and thank Ronald Parr for insightful advice. This research was supported in part by DARPA, DOE, NIH, ONR and NSF.  ... 
arXiv:1808.03030v1 fatcat:i3swiw5wrvdnnk7nry6ijir4rm

On Weak Super Ricci Flow through Neckpinch [article]

Sajjad Lakzian, Michael Munn
2020 arXiv   pre-print
We also show the spacetime is a refined weak super Ricci flow if and only if the flow is a smooth Ricci flow with possibly singular final time.  ...  We introduce the notion of a Ricci flow metric measure spacetime and of a weak (refined) super Ricci flow associated to convex cost functions (cost functions which are increasing convex functions of the  ...  The authors would also like to thank the Hausdorff Research Institute where some of this work was completed during a Trimester program in Optimal Transport. SL is grateful to D.  ... 
arXiv:2008.10508v1 fatcat:5v5s23vvhzd55ivyhbxrqzbvui

Optimally Reliable Cheap Payment Flows on the Lightning Network [article]

Rene Pickhardt, Stefan Richter
2021 arXiv   pre-print
) integer minimum cost flow with a separable and convex cost function.  ...  This algorithm works by updating the probability distributions with the information gained from both successful and unsuccessful paths on prior rounds.  ...  Acknowledgements This research was partially sponsored by the Norwegian University of Science and Technology (NTNU).  ... 
arXiv:2107.05322v1 fatcat:ffhwscsyxrevtb4l3clutc3r3y

Heterogeneous gradient flows in the topology of fibered optimal transport [article]

Jan Peszek, David Poyato
2022 arXiv   pre-print
We introduce an optimal transport topology on the space of probability measures over a fiber bundle, which penalizes the transport cost from one fiber to another.  ...  Optimal transport becomes then constrained to happen along fixed fibers.  ...  Section 2 is dedicated to a review of classical optimal transport theory and the theory of random probability measures.  ... 
arXiv:2203.08104v2 fatcat:kzgn7a73v5alrin7ml676mvsve

Mean field Variational Inference via Wasserstein Gradient Flow [article]

Rentian Yao, Yun Yang
2022 arXiv   pre-print
Our theoretical analysis relies on optimal transport theory and subdifferential calculus in the space of probability measures.  ...  In this work, we develop a general computational framework for implementing MF-VI via Wasserstein gradient flow (WGF), a gradient flow over the space of probability measures.  ...  probability distributions with finite second moments endowed with the 2-Wasserstein metric W 2 [3] .  ... 
arXiv:2207.08074v1 fatcat:su36yn5y6ffhhbr5qshx6iqnzu
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