Multimarginal Optimal Transport with a Tree-Structured Cost and the Schrödinger Bridge Problem

AbstractThis paper exploit the equivalence between the Schrödinger Bridge problem (Léonard in J Funct Anal 262:1879–1920, 2012; Nelson in Phys Rev 150:1079, 1966; Schrödinger in Über die umkehrung der naturgesetze. Verlag Akademie der wissenschaften in kommission bei Walter de Gruyter u, Company, 1931) and the entropy penalized optimal transport (Cuturi in: Advances in neural information processing systems, pp 2292–2300, 2013; Galichon and Salanié in: Matching with trade-offs: revealed preferences over competing characteristics. CEPR discussion paper no. DP7858, 2010) in order to find a different approach to the duality, in the spirit of optimal transport. This approach results in a priori estimates which are consistent in the limit when the regularization parameter goes to zero. In particular, we find a new proof of the existence of maximizing entropic-potentials and therefore, the existence of a solution of the Schrödinger system. Our method extends also when we have more than two marginals: the main new result is the proof that the Sinkhorn algorithm converges even in the continuous multi-marginal case. This provides also an alternative proof of the convergence of the Sinkhorn algorithm in two marginals.

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Hopf–Cole transformation via generalized Schrödinger bridge problem

Journal of Differential Equations ◽

10.1016/j.jde.2020.10.029 ◽

2021 ◽

Vol 274 ◽

pp. 788-827 ◽

Cited By ~ 1

Author(s):

Flavien Léger ◽

Wuchen Li

Keyword(s):

Bridge Problem ◽

Schrödinger Bridge

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Solving Schrödinger Bridges via Maximum Likelihood

Entropy ◽

10.3390/e23091134 ◽

2021 ◽

Vol 23 (9) ◽

pp. 1134

Author(s):

Francisco Vargas ◽

Pierre Thodoroff ◽

Austen Lamacraft ◽

Neil Lawrence

Keyword(s):

Machine Learning ◽

Maximum Likelihood ◽

Probability Distributions ◽

Numerical Procedure ◽

Likelihood Estimation ◽

Machine Learning Techniques ◽

Stochastic Evolution ◽

Learning Techniques ◽

Bridge Problem ◽

Schrödinger Bridge

The Schrödinger bridge problem (SBP) finds the most likely stochastic evolution between two probability distributions given a prior stochastic evolution. As well as applications in the natural sciences, problems of this kind have important applications in machine learning such as dataset alignment and hypothesis testing. Whilst the theory behind this problem is relatively mature, scalable numerical recipes to estimate the Schrödinger bridge remain an active area of research. Our main contribution is the proof of equivalence between solving the SBP and an autoregressive maximum likelihood estimation objective. This formulation circumvents many of the challenges of density estimation and enables direct application of successful machine learning techniques. We propose a numerical procedure to estimate SBPs using Gaussian process and demonstrate the practical usage of our approach in numerical simulations and experiments.

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Optimal transport and unitary orbits in C⁎-algebras

Journal of Functional Analysis ◽

10.1016/j.jfa.2021.109068 ◽

2021 ◽

Vol 281 (5) ◽

pp. 109068

Author(s):

Bhishan Jacelon ◽

Karen R. Strung ◽

Alessandro Vignati

Keyword(s):

Optimal Transport ◽

C Algebras

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Modeling Epidemic Spread among a Commuting Population Using Transport Schemes

Mathematics ◽

10.3390/math9161861 ◽

2021 ◽

Vol 9 (16) ◽

pp. 1861

Author(s):

Daniela Calvetti ◽

Alexander P. Hoover ◽

Johnie Rose ◽

Erkki Somersalo

Keyword(s):

Network Model ◽

Transport Theory ◽

Optimal Transport ◽

State Level ◽

Compartment Model ◽

Census Bureau ◽

Mitigation Strategies ◽

Mitigation Measures ◽

Metapopulation Model ◽

Infection Dynamics

Understanding the dynamics of the spread of COVID-19 between connected communities is fundamental in planning appropriate mitigation measures. To that end, we propose and analyze a novel metapopulation network model, particularly suitable for modeling commuter traffic patterns, that takes into account the connectivity between a heterogeneous set of communities, each with its own infection dynamics. In the novel metapopulation model that we propose here, transport schemes developed in optimal transport theory provide an efficient and easily implementable way of describing the temporary population redistribution due to traffic, such as the daily commuter traffic between work and residence. Locally, infection dynamics in individual communities are described in terms of a susceptible-exposed-infected-recovered (SEIR) compartment model, modified to account for the specific features of COVID-19, most notably its spread by asymptomatic and presymptomatic infected individuals. The mathematical foundation of our metapopulation network model is akin to a transport scheme between two population distributions, namely the residential distribution and the workplace distribution, whose interface can be inferred from commuter mobility data made available by the US Census Bureau. We use the proposed metapopulation model to test the dynamics of the spread of COVID-19 on two networks, a smaller one comprising 7 counties in the Greater Cleveland area in Ohio, and a larger one consisting of 74 counties in the Pittsburgh–Cleveland–Detroit corridor following the Lake Erie’s American coastline. The model simulations indicate that densely populated regions effectively act as amplifiers of the infection for the surrounding, less densely populated areas, in agreement with the pattern of infections observed in the course of the COVID-19 pandemic. Computed examples show that the model can be used also to test different mitigation strategies, including one based on state-level travel restrictions, another on county level triggered social distancing, as well as a combination of the two.

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Quantum Optimal Transport with Quantum Channels

Annales Henri Poincaré ◽

10.1007/s00023-021-01042-3 ◽

2021 ◽

Author(s):

Giacomo De Palma ◽

Dario Trevisan

Keyword(s):

Optimal Transport ◽

Quantum Channels

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279 WACSAW: A Statistical and Individually Adaptive Method to Determine Sleep and Wakefulness from Activity

SLEEP ◽

10.1093/sleep/zsab072.278 ◽

2021 ◽

Vol 44 (Supplement_2) ◽

pp. A111-A112

Author(s):

Austin Vandegriffe ◽

V A Samaranayake ◽

Matthew Thimgan

Keyword(s):

Time Series ◽

Optimal Transport ◽

Activity Patterns ◽

Adaptive Method ◽

Wasserstein Distance ◽

Activity Data ◽

Commercial Grade ◽

Movement Data ◽

The Difference ◽

Levene Test

Abstract Introduction Technological innovations have broadened the type and amount of activity data that can be captured in the home and under normal living conditions. Yet, converting naturalistic activity patterns into sleep and wakefulness states has remained a challenge. Despite the successes of current algorithms, they do not fill all actigraphy needs. We have developed a novel statistical approach to determine sleep and wakefulness times, called the Wasserstein Algorithm for Classifying Sleep and Wakefulness (WACSAW), and validated the algorithm in a small cohort of healthy participants. Methods WACSAW functional routines: 1) Conversion of the triaxial movement data into a univariate time series; 2) Construction of a Wasserstein weighted sum (WSS) time series by measuring the Wasserstein distance between equidistant distributions of movement data before and after the time-point of interest; 3) Segmenting the time series by identifying changepoints based on the behavior of the WSS series; 4) Merging segments deemed similar by the Levene test; 5) Comparing segments by optimal transport methodology to determine the difference from a flat, invariant distribution at zero. The resulting histogram can be used to determine sleep and wakefulness parameters around a threshold determined for each individual based on histogram properties. To validate the algorithm, participants wore the GENEActiv and a commercial grade actigraphy watch for 48 hours. The accuracy of WACSAW was compared to a detailed activity log and benchmarked against the results of the output from commercial wrist actigraph. Results WACSAW performed with an average accuracy, sensitivity, and specificity of >95% compared to detailed activity logs in 10 healthy-sleeping individuals of mixed sexes and ages. We then compared WACSAW’s performance against a common wrist-worn, commercial sleep monitor. WACSAW outperformed the commercial grade system in each participant compared to activity logs and the variability between subjects was cut substantially. Conclusion The performance of WACSAW demonstrates good results in a small test cohort. In addition, WACSAW is 1) open-source, 2) individually adaptive, 3) indicates individual reliability, 4) based on the activity data stream, and 5) requires little human intervention. WACSAW is worthy of validating against polysomnography and in patients with sleep disorders to determine its overall effectiveness. Support (if any):

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