Computations of Optimal Transport Distance with Fisher Information Regularization

Li, Wuchen; Yin, Pengbin; Osher, Stanley

doi:10.1007/s10915-017-0599-0

Cited by 50 publications

(45 citation statements)

References 29 publications

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“…As we did before, we assume U = 0 for simplicity. Differentiating the relation (38) and using the definition of Hessian we get…”

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confidence: 99%

A second order equation for Schrödinger bridges with applications to the hot gas experiment and entropic transportation cost

Conforti

2018

Probab. Theory Relat. Fields

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The Schrödinger problem is obtained by replacing the mean square distance with the relative entropy in the Monge-Kantorovich problem. It was first addressed by Schrödinger as the problem of describing the most likely evolution of a large number of Brownian particles conditioned to reach an "unexpected configuration". Its optimal value, the entropic transportation cost, and its optimal solution, the Schrödinger bridge, stand as the natural probabilistic counterparts to the transportation cost and displacement interpolation. Moreover, they provide a natural way of lifting from the point to the measure setting the concept of Brownian bridge. In this article, we prove that the Schrödinger bridge solves a second order equation in the Riemannian structure of optimal transport. Roughly speaking, the equation says that its acceleration is the gradient of the Fisher information. Using this result, we obtain a fine quantitative description of the dynamics, and a new functional inequality for the entropic transportation cost, that generalize Talagrand's transportation inequality. Finally, we study the convexity of the Fisher information along Schrödigner bridges, under the hypothesis that the associated reciprocal characteristic is convex. The techniques developed in this article are also well suited to study the Feynman-Kac penalisations of Brownian motion. 1 arXiv:1704.04821v4 [math.PR]

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“…As we did before, we assume U = 0 for simplicity. Differentiating the relation (38) and using the definition of Hessian we get…”

mentioning

confidence: 99%

A second order equation for Schrödinger bridges with applications to the hot gas experiment and entropic transportation cost

Conforti

2018

Probab. Theory Relat. Fields

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“…Since the work of Mikami, Thieullen, Leonard, Cuturi [47], [48], [49], [43], [44], [26], a large number of papers have appeared where Schrödinger bridge problems are viewed as regularization of the important Optimal Mass Transport (OMT) problem, see e.g., [8], [17], [18], [19], [45], [2], [22]. This is, of course, interesting and extremely effective as OMT is computationally challenging [3], [7].…”

Section: Final Commentsmentioning

confidence: 99%

Relaxed Schrödinger Bridges and Robust Network Routing

Chen

Georgiou

Pavon

et al. 2020

IEEE Trans. Control Netw. Syst.

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We consider network routing under random link failures with a desired final distribution. We provide a mathematical formulation of a relaxed transport problem where the final distribution only needs to be close to the desired one. The problem is a maximum entropy problem for path distributions with an extra terminal cost. We show that the unique solution may be obtained solving a generalized Schrödinger system. An iterative algorithm to compute the solution is provided. It contracts the Hilbert metric with contraction ratio less than 1/2 leading to extremely fast convergence.

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“…Various aspects of regularized OMT have been studied in a number of recent works including [25,26,27,28] which have extensive lists of references.…”

Section: Optimal Mass Transport With Diffusionmentioning

confidence: 99%

Optimal Mass Transport Kinetic Modeling for Head and Neck DCE-MRI: Initial Analysis

Elkin

Nadeem

LoCastro

et al. 2019

Preprint

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Current state-of-the-art models for estimating the pharmacokinetic parameters do not account for intervoxel movement of the contrast agent (CA).We introduce an optimal mass transport (OMT) formulation that naturally handles intervoxel CA movement and distinguishes between advective and diffusive flows. Ten patients with head and neck squamous cell carcinoma (HNSCC) were enrolled in the study between June 2014 and October 2015 and underwent DCE MRI imaging prior to beginning treatment. The CA tissue concentration information was taken as the input in the data-driven OMT model. The OMT approach was tested on HNSCC DCE data that provides quantitative information for forward flux (ΦF ) and backward flux (ΦB ). OMT-derived ΦF was compared with the volume transfer constant for CA, K t r ans , derived from the Extended Tofts Model (ETM). The OMT-derived flows showed a consistent jump in the CA diffusive behavior across the images in accordance with the known CA dynamics. The mean forward flux was 0.0082 ± 0.0091 (min −1 ) whereas the mean advective component was 0.0052±0.0086 (min −1 ) in the HNSCC patients. The diffusive percentages in forward and backward flux ranged from 8. 67-18.76% and 12.76-30.36%, respectively. The OMT model accounts for intervoxel CA movement and results show that the forward flux (ΦF ) is comparable with the ETMderived K t r ans . This is a novel data-driven study based on optimal mass transport principles applied to patient DCE imaging to analyze CA flow in HNSCC.

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Computations of Optimal Transport Distance with Fisher Information Regularization

Cited by 50 publications

References 29 publications

A second order equation for Schrödinger bridges with applications to the hot gas experiment and entropic transportation cost

A second order equation for Schrödinger bridges with applications to the hot gas experiment and entropic transportation cost

Relaxed Schrödinger Bridges and Robust Network Routing

Optimal Mass Transport Kinetic Modeling for Head and Neck DCE-MRI: Initial Analysis

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