Inverse Optimal Transport

Stuart, Andrew M.; Wolfram, Marie-Thérèse

doi:10.1137/19m1261122

Cited by 10 publications

(7 citation statements)

References 27 publications

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“…Studying the inverse problem and learning the conformational energy landscape: Ultimately, understanding the mechanisms explaining the dynamics between different conformational states can be cast as an inverse problem, aiming to recover the inner cost function associated with observed trajectories between two shapes/conformations. This inverse problem has been an active subject of research for the past few years, with some hypotheses made on the form of the cost function [ 34 - 36 ], and various methods, from deep learning [ 37 ] to Bayesian MCMC methods [ 38 ]. Those methods have in common that they heavily rely on the computation of many forward OT solutions, to learn the cost function.…”

Section: Discussionmentioning

confidence: 99%

Application of transport-based metric for continuous interpolation between cryo-EM density maps

Ecoffet¹,

Woollard²,

Kushner³

et al. 2021

MATH

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<abstract><p>Cryogenic electron microscopy (cryo-EM) has become widely used for the past few years in structural biology, to collect single images of macromolecules "frozen in time". As this technique facilitates the identification of multiple conformational states adopted by the same molecule, a direct product of it is a set of 3D volumes, also called EM maps. To gain more insights on the possible mechanisms that govern transitions between different states, and hence the mode of action of a molecule, we recently introduced a bioinformatic tool that interpolates and generates morphing trajectories joining two given EM maps. This tool is based on recent advances made in optimal transport, that allow efficient evaluation of Wasserstein barycenters of 3D shapes. As the overall performance of the method depends on various key parameters, including the sensitivity of the regularization parameter, we performed various numerical experiments to demonstrate how MorphOT can be applied in different contexts and settings. Finally, we discuss current limitations and further potential connections between other optimal transport theories and the conformational heterogeneity problem inherent with cryo-EM data.</p></abstract>

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Section: Discussionmentioning

confidence: 99%

Application of transport-based metric for continuous interpolation between cryo-EM density maps

Ecoffet¹,

Woollard²,

Kushner³

et al. 2021

MATH

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“…Here the MFG system describes the dynamics of agents, where we have observations about the motion and strategy adopted by the agents during the game. Our observation is time dependent, which is different from the static joint distribution in [30,37]. In addition, for MFG with interaction energy, MFG dynamics can not be formulated as a minimizer of linear programming.…”

Section: Related Workmentioning

confidence: 91%

“…[37] proposes a framework to learn the unknown ground costs from noisy observations during optimal transport. In particular, [30,37] focus on the linear programming formulation of inverse optimal transport problems. Compared to existing works, we focus on PDE formulations of MFGs.…”

Section: Related Workmentioning

confidence: 99%

“…For example, [30] proposes a unified data-driven framework to learn the adaptive, nonlinear interaction cost functions in the matching process from data corrupted by noise and then make predictions to new matchings. [37] proposes a framework to learn the unknown ground costs from noisy observations during optimal transport. In particular, [30,37] focus on the linear programming formulation of inverse optimal transport problems.…”

Section: Related Workmentioning

confidence: 99%

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A mean field game inverse problem

Ding¹,

Li²,

Osher³

et al. 2020

Preprint

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Mean-field games arise in various fields including economics, engineering and machine learning. They study strategic decision making in large populations where the individuals interact via certain mean-field quantities. The ground metrics and running costs of the games are of essential importance but are often unknown or only partially known. In this paper, we propose mean-field game inverse-problem models to reconstruct the ground metrics and interaction kernels in the running costs. The observations are the macro motions, to be specific, the density distribution and velocity field of the agents. They can be corrupted by noise to some extent. Our models are PDE constrained optimization problems, which are solvable by first-order primal-dual methods. Besides, we apply Bregman iterations to find the optimal model parameters. We numerically demonstrate that our model is both efficient and robust to noise.

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“…Implementation of inverse optimization have been applied in various fields , where it is used to measure operational variance in transit operators, to detect shifts in travel/traffic objectives in system security and risk management, to learn mechanism in autonomous vehicles and so forth. Andrew et al [24] present a systematic method to derive obscure costs from observations with noisy data of the optimal transportation plans. They implement a formulation of the problem based on graph theory, where nodes represent countries of graphs and assign nonzero weight on the edges between adjacent countries which have a common border.…”

Section: Introductionmentioning

confidence: 99%

Stability of Solutions for Parametric Inverse Nonlinear Cost Transportation Problem

Mousa

Abo-Elnaga

2020

Mathematics

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This paper investigates the solution for an inverse of a parametric nonlinear transportation problem, in which, for a certain values of the parameters, the cost of the unit transportation in the basic problem are adapted as little as possible so that the specific feasible alternative become an optimal solution. In addition, a solution stability set of these parameters was investigated to keep the new optimal solution (feasible one) is unchanged. The idea of this study based on using a tuning parameters λ∈Rm in the function of the objective and input parameters υ∈Rl in the set of constraint. The inverse parametric nonlinear cost transportation problem P(λ,υ), where the tuning parameters λ∈Rm in the objective function are tuned (adapted) as less as possible so that the specific feasible solution x∘ has been became the optimal ones for a certain values of υ∈Rl, then, a solution stability set of the parameters was investigated to keep the new optimal solution x∘ unchanged. The proposed method consists of three phases. Firstly, based on the optimality conditions, the parameter λ∈Rm are tuned as less as possible so that the initial feasible solution x∘ has been became new optimal solution. Secondly, using input parameters υ∈Rl resulting problem is reformulated in parametric form P(υ). Finally, based on the stability notions, the availability domain of the input parameters was detected to keep its optimal solution unchanged. Finally, to clarify the effectiveness of the proposed algorithm not only for the inverse transportation problems but also, for the nonlinear programming problems; numerical examples treating the inverse nonlinear programming problem and the inverse transportation problem of minimizing the nonlinear cost functions are presented.

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Inverse Optimal Transport

Cited by 10 publications

References 27 publications

Application of transport-based metric for continuous interpolation between cryo-EM density maps

Application of transport-based metric for continuous interpolation between cryo-EM density maps

A mean field game inverse problem

Stability of Solutions for Parametric Inverse Nonlinear Cost Transportation Problem

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