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On the Kähler Geometry of Certain Optimal Transport Problems [article]

Gabriel Khan, Jun Zhang
2019 arXiv   pre-print
Let X and Y be domains of R^n equipped with respective probability measures μ and ν. We consider the problem of optimal transport from μ to ν with respect to a cost function c: X × Y →R.  ...  Taken together, these results provide a geometric framework for optimal transport which is complementary to the pseudo-Riemannian theory of Kim and McCann.  ...  In Section 2 we discuss some background information on optimal transport. Section 3 discusses some background information on Hessian manifolds and the curvature of the Sasaki metric.  ... 
arXiv:1812.00032v4 fatcat:ocuwnaaacrhe5giog23wlrkece

Kähler-Ricci Flow preserves negative anti-bisectional curvature [article]

Gabriel Khan, Fangyang Zheng
2021 arXiv   pre-print
Anal. 2 (2020), 397-426), the first named author and J. Zhang found a connection between the regularity theory of optimal transport and the curvature of Kähler manifolds.  ...  We provide several applications of these results – in complex geometry, optimal transport, and affine geometry.  ...  Optimal transport and the role of anti-bisectional curvature. Optimal transport studies the most economical way to transport resources.  ... 
arXiv:2011.07181v2 fatcat:ggz6vgkzujbkhb5wqw3s7lv5ne

Parabolic Optimal Transport Equations on Manifolds

Young-Heon Kim, Jeffrey Streets, Micah Warren
2011 International mathematics research notices  
Young-Heon Kim, Robert McCann and Micah Warren. Calibrating optimal transportation with Pseudo-Riemannian geometry. Math. Res. Lett. 17, (6) (2010) 1183-1197. 21.  ...  Young-Heon Kim, Jeffrey Streets and Micah Warren. Parabolic optimal transport equations on manifolds. Int. Math. Res. Notices 2011, doi = 10.1093/imrn/rnr188. 22. Simon Brendle and Micah Warren.  ... 
doi:10.1093/imrn/rnr188 fatcat:aqbpjpc4dffwtiqmwriebrooqy

A Smoothness Energy without Boundary Distortion for Curved Surfaces [article]

Oded Stein, Alec Jacobson, Max Wardetzky, Eitan Grinspun
2019 arXiv   pre-print
We also provide an implementation that enables the use of the Hessian energy for applications on curved surfaces for which current quadratic smoothness energies do not produce satisfying results, and observe  ...  This energy features the curved Hessian of functions on manifolds as well as an additional curvature term which results from applying the Weitzenbock identity.  ...  cross lilium, hand, puppet head by Cosmic blobs), the Stanford 3D Scanning Repository [Graphics 2014 ] (armadillo), Crane [2018] (rubber duck, man-bridge, spot the cow, Nefertiti by Nora Al-Badri and  ... 
arXiv:1905.09777v1 fatcat:57vhwvn3ffhepbfoty6unpcg6q

The geometry of positively curved Kähler metrics on tube domains [article]

Gabriel Khan, Jun Zhang, Fangyang Zheng
2020 arXiv   pre-print
Finally, we discuss how the interplay between optimal transport and complex geometry can be used to define a "synthetic" version of curvature bounds for Kähler manifolds whose associated potential is merely  ...  These examples are also of interest to optimal transport, as they can be used to generate new examples of cost functions which only depend on the Euclidean distance between points and satisfy the weak  ...  These dual coordinates play an essential role in the study of Hessian manifolds, and also in the optimal transport of Ψ-costs.  ... 
arXiv:2001.06155v1 fatcat:gizl7go7yzfwfhc3iqswz47hiu

Mean field information Hessian matrices on graphs [article]

Wuchen Li, Linyuan Lu
2022 arXiv   pre-print
And the "mean-field" means nonlinear weight functions of probabilities supported on graphs. These two concepts define a mean-field optimal transport type metric.  ...  We name their smallest eigenvalues as mean-field Ricci curvature bounds on graphs. We next provide examples on two-point spaces and graph products.  ...  And the constant κ is the proposed mean-field Ricci curvature lower bound. 5.5. Hessian operators in optimal transport spaces.  ... 
arXiv:2203.06307v1 fatcat:foqn6hfe6jg5pj3x3jcn7iltka

Surface-based shape classification using Wasserstein distance

Ming Ma, Na Lei, Kehua Su, Junwei Zhang, Chengfeng Wen, Li Cui, Xin Fan, Xianfeng Gu
2015 Geometry Imaging and Computing  
The transportation cost of this optimal mass transport defines the Wasserstein distance between two surfaces, which intrinsically measures the dissimilarities between surface based shapes and thus can  ...  Given any two probability measures on two surfaces, our method is capable of obtaining a unique optimal mass transport map between them.  ...  Optimal mass transport Monge [6] raised the optimal mass transport problem in the 18th century. Definition 3.1 (Optimal Mass Transport).  ... 
doi:10.4310/gic.2015.v2.n4.a1 fatcat:y5wa6lxxhvfzxipilnwliuzicm

Self-Tuning Stochastic Optimization with Curvature-Aware Gradient Filtering [article]

Ricky T. Q. Chen, Dami Choi, Lukas Balles, David Duvenaud, Philipp Hennig
2020 arXiv   pre-print
Based on this intuition, we explore the use of exact per-sample Hessian-vector products and gradients to construct optimizers that are self-tuning and hyperparameter-free.  ...  Standard first-order stochastic optimization algorithms base their updates solely on the average mini-batch gradient, and it has been shown that tracking additional quantities such as the curvature can  ...  In contrast, we focus on explicitly transporting via the full Hessian.  ... 
arXiv:2011.04803v1 fatcat:pek6vnmdfrg77hphsfxgg56f4e

On the c-concavity with respect to the quadratic cost on a manifold [article]

Federico Glaudo
2018 arXiv   pre-print
From this, we deduce a sufficient condition for the optimality of transport maps.  ...  Introduction Let us briefly recall the optimal transport problem on R n with quadratic cost c(x, y) = 1 2 d 2 (x, y).  ...  Furthermore, in positive curvature, the Hessian of the square of the distance can be strictly smaller than the metric and consequently we will need to ask a stricter condition on the Hessian of the function  ... 
arXiv:1802.06366v1 fatcat:5cfdhdlvevh4bovydybepikpeu

Ricci curvature for parametric statistics via optimal transport [article]

Wuchen Li, Guido Montufar
2020 arXiv   pre-print
We discuss examples of Ricci curvature lower bounds and convergence rates in exponential family models.  ...  Within these definitions, the Ricci curvature is related to both, information geometry and Wasserstein geometry.  ...  In recent years, optimal transport contributed a viewpoint that connects Ricci curvature and information functionals.  ... 
arXiv:1807.07095v3 fatcat:z7lzzaluwzgkbmr6h4owtddv5y

Curvature-Dependant Global Convergence Rates for Optimization on Manifolds of Bounded Geometry [article]

Mario Lezcano-Casado
2020 arXiv   pre-print
We give curvature-dependant convergence rates for the optimization of weakly convex functions defined on a manifold of 1-bounded geometry via Riemannian gradient descent and via the dynamic trivialization  ...  We compute these bounds explicitly for some manifolds commonly used in the optimization literature such as the special orthogonal group and the real Grassmannian.  ...  Raphael Hauser and Prof. Coralia Cartis for useful feedback and pointers in the early stages of this project and encouragement to follow the line of work that led to this final version.  ... 
arXiv:2008.02517v1 fatcat:sbvrbf2gpbas7noyf2z5lxfspu

Accelerated Stochastic Quasi-Newton Optimization on Riemann Manifolds [article]

Anirban Roychowdhury
2017 arXiv   pre-print
We discuss a couple of ways to obtain the correction pairs used to calculate the product of the gradient with the inverse Hessian, and empirically demonstrate their use in synthetic experiments on computation  ...  We provide a new convergence proof for strongly convex functions without using curvature conditions on the manifold, as well as a convergence discussion for nonconvex functions.  ...  The curvature information provided by the Hessian estimate allows superlinear convergence in ideal settings [12] .  ... 
arXiv:1704.01700v3 fatcat:6mb7mq2bqvdlbjtigqr625g7x4

A Riemannian interpolation inequality à la Borell, Brascamp and Lieb

Dario Cordero-Erausquin, Robert J. McCann, Michael Schmuckenschläger
2001 Inventiones Mathematicae  
The method uses optimal mappings from mass transportation theory. Along the way, several new properties are established for optimal mass transport and interpolating maps on a Riemannian manifold.  ...  Due to the curvature of the manifold, the new Riemannian versions of these theorems incorporate a volume distortion factor which can, however, be controlled via lower bounds on Ricci curvature.  ...  As in [27] , The map F may be referred to either as the optimal map or optimal mass transport between µ and ν.  ... 
doi:10.1007/s002220100160 fatcat:sdqewnxzjrayhgbwgtfjao5rwe

Variational Wasserstein Clustering [article]

Liang Mi, Wen Zhang, Xianfeng Gu, Yalin Wang
2018 arXiv   pre-print
We propose a new clustering method based on optimal transportation.  ...  We solve optimal transportation with variational principles, and investigate the use of power diagrams as transportation plans for aggregating arbitrary domains into a fixed number of clusters.  ...  Acknowledgemants The research is partially supported by National Institutes of Health (R21AG043760, RF1AG051710, and R01EB025032), and National Science Foundation (DMS-1413417 and IIS-1421165).  ... 
arXiv:1806.09045v4 fatcat:t3tuvhi4jnczzohwqgahfvnbmm

A Remark on the Potentials of Optimal Transport Maps [article]

Paul W.Y. Lee
2010 arXiv   pre-print
Optimal maps, solutions to the optimal transportation problems, are completely determined by the corresponding c-convex potential functions.  ...  introduction to the theory of optimal transportation problem).  ...  Then the optimal transportation problem corresponding to the transportation cost (2) has a unique solution ϕ : M → M.  ... 
arXiv:1006.3917v1 fatcat:bmbxihlcljbbhe4rg4rumxibrq
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