scholarly journals Fused Gromov-Wasserstein Distance for Structured Objects

Algorithms ◽  
2020 ◽  
Vol 13 (9) ◽  
pp. 212
Author(s):  
Titouan Vayer ◽  
Laetitia Chapel ◽  
Remi Flamary ◽  
Romain Tavenard ◽  
Nicolas Courty
Keyword(s):  
Machine Learning ◽  
Transport Theory ◽  
Optimal Transport ◽  
Wasserstein Distance ◽  
Learning Objects ◽  
Mathematical Framework ◽  
Structure Information ◽  
Finite Samples ◽  
Concentration Result ◽  
Structured Objects

Optimal transport theory has recently found many applications in machine learning thanks to its capacity to meaningfully compare various machine learning objects that are viewed as distributions. The Kantorovitch formulation, leading to the Wasserstein distance, focuses on the features of the elements of the objects, but treats them independently, whereas the Gromov–Wasserstein distance focuses on the relations between the elements, depicting the structure of the object, yet discarding its features. In this paper, we study the Fused Gromov-Wasserstein distance that extends the Wasserstein and Gromov–Wasserstein distances in order to encode simultaneously both the feature and structure information. We provide the mathematical framework for this distance in the continuous setting, prove its metric and interpolation properties, and provide a concentration result for the convergence of finite samples. We also illustrate and interpret its use in various applications, where structured objects are involved.

scholarly journals Detecting Changes in Forced Climate Attractors with Wasserstein Distance

2017 ◽  
Author(s):  
Yoann Robin ◽  
Pascal Yiou ◽  
Philippe Naveau
Keyword(s):  
Transport Theory ◽  
Optimal Transport ◽  
Warming Trend ◽  
Wasserstein Distance ◽  
Mean Values ◽  
External Forcings ◽  
Attractor Dynamics ◽  
Important Challenge ◽  
Natural Building ◽  
Optimal Transport Theory

Abstract. The climate system can been described by a dynamical system and its associated attractor. The dynamics of this attractor depends on the external forcings that influence the climate. Such forcings can affect the mean values or variances, but regions of the attractor that are seldom visited can also be affected. It is an important challenge to measure how the climate attractor responds to different forcings. Currently, the Euclidean distance or similar measures like the Mahalanobis distance have been favoured to measure discrepancies between two climatic situations. Those distances do not have a natural building mechanism to take into account the attractor dynamics. In this paper, we argue that a Wasserstein distance, stemming from optimal transport theory, offers an efficient and practical way to discriminate between dynamical systems. After treating a toy example, we explore how the Wasserstein distance can be applied and interpreted to detect non-autonomous dynamics from a Lorenz system driven by seasonal cycles and a warming trend.

scholarly journals Optimal transport for variational data assimilation

2018 ◽  
Vol 25 (1) ◽  
pp. 55-66 ◽  
Author(s):  
Nelson Feyeux ◽  
Arthur Vidard ◽  
Maëlle Nodet
Keyword(s):  
Data Assimilation ◽  
Transport Theory ◽  
Optimal Transport ◽  
Wasserstein Distance ◽  
Variational Data Assimilation ◽  
Water Model ◽  
Position Errors ◽  
Optimal Transport Theory ◽  
The Cost ◽  
Model Trajectory

Abstract. Usually data assimilation methods evaluate observation-model misfits using weighted L2 distances. However, it is not well suited when observed features are present in the model with position error. In this context, the Wasserstein distance stemming from optimal transport theory is more relevant.This paper proposes the adaptation of variational data assimilation for the use of such a measure. It provides a short introduction of optimal transport theory and discusses the importance of a proper choice of scalar product to compute the cost function gradient. It also extends the discussion to the way the descent is performed within the minimization process.These algorithmic changes are tested on a nonlinear shallow-water model, leading to the conclusion that optimal transport-based data assimilation seems to be promising to capture position errors in the model trajectory.

scholarly journals Unsupervised collaborative learning based on Optimal Transport theory

2021 ◽  
Vol 30 (1) ◽  
pp. 698-719
Author(s):  
Fatima-Ezzahraa Ben-Bouazza ◽  
Younès Bennani ◽  
Guénaël Cabanes ◽  
Abdelfettah Touzani
Keyword(s):  
Collaborative Learning ◽  
Transport Theory ◽  
Optimal Transport ◽  
Wasserstein Distance ◽  
Data Sets ◽  
Multiple Data ◽  
Novel Approach ◽  
Multiple Data Sets ◽  
Optimal Transport Theory ◽  
The Right

Abstract Collaborative learning has recently achieved very significant results. It still suffers, however, from several issues, including the type of information that needs to be exchanged, the criteria for stopping and how to choose the right collaborators. We aim in this paper to improve the quality of the collaboration and to resolve these issues via a novel approach inspired by Optimal Transport theory. More specifically, the objective function for the exchange of information is based on the Wasserstein distance, with a bidirectional transport of information between collaborators. This formulation allows to learns a stopping criterion and provide a criterion to choose the best collaborators. Extensive experiments are conducted on multiple data-sets to evaluate the proposed approach.

scholarly journals Detecting changes in forced climate attractors with Wasserstein distance

2017 ◽  
Vol 24 (3) ◽  
pp. 393-405 ◽  
Author(s):  
Yoann Robin ◽  
Pascal Yiou ◽  
Philippe Naveau
Keyword(s):  
Transport Theory ◽  
Optimal Transport ◽  
Warming Trend ◽  
Wasserstein Distance ◽  
Mean Values ◽  
External Forcings ◽  
Attractor Dynamics ◽  
Important Challenge ◽  
Natural Building ◽  
Optimal Transport Theory

Abstract. The climate system can been described by a dynamical system and its associated attractor. The dynamics of this attractor depends on the external forcings that influence the climate. Such forcings can affect the mean values or variances, but regions of the attractor that are seldom visited can also be affected. It is an important challenge to measure how the climate attractor responds to different forcings. Currently, the Euclidean distance or similar measures like the Mahalanobis distance have been favored to measure discrepancies between two climatic situations. Those distances do not have a natural building mechanism to take into account the attractor dynamics. In this paper, we argue that a Wasserstein distance, stemming from optimal transport theory, offers an efficient and practical way to discriminate between dynamical systems. After treating a toy example, we explore how the Wasserstein distance can be applied and interpreted to detect non-autonomous dynamics from a Lorenz system driven by seasonal cycles and a warming trend.

scholarly journals Optimal Transport for Variational Data Assimilation

2017 ◽  
Author(s):  
Nelson Feyeux ◽  
Arthur Vidard ◽  
Maëlle Nodet
Keyword(s):  
Data Assimilation ◽  
Transport Theory ◽  
Optimal Transport ◽  
Wasserstein Distance ◽  
Variational Data Assimilation ◽  
Initial Condition ◽  
Geometrical Properties ◽  
Euclidean Distances ◽  
Optimal Transport Theory ◽  
The Impact

Abstract. Variational data assimilation methods are designed to estimate an unknown initial condition of a model using observations. To do so, one needs to compare model outputs and observations. This is generally performed using Euclidean distances. This paper investigates another distance choice: the Wasserstein distance, stemming from optimal transport theory. We develop a variational data assimilation method using this distance. We investigate the impact of the scalar product, the 5 gradient choice as well as the minimization algorithm. With appropriate choices, we show successful results on preliminary experiments. Optimal-transport-based optimization seems to be promising to preserve the geometrical properties of the estimated initial condition.

scholarly journals Modeling Epidemic Spread among a Commuting Population Using Transport Schemes

Mathematics ◽  
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.

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

SLEEP ◽  
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):

scholarly journals Wildlife Monitoring Using a Multi-UAV System with Optimal Transport Theory

2021 ◽  
Vol 11 (9) ◽  
pp. 4070
Author(s):  
Rabiul Hasan Kabir ◽  
Kooktae Lee
Keyword(s):  
Detection Rate ◽  
Transport Theory ◽  
Optimal Transport ◽  
Position Estimation ◽  
Wildlife Monitoring ◽  
High Detection Rate ◽  
Monitoring Strategies ◽  
Spatio Temporal ◽  
Optimal Transport Theory ◽  
Multi Uav

This paper addresses a wildlife monitoring problem using a team of unmanned aerial vehicles (UAVs) with the optimal transport theory. The state-of-the-art technology using UAVs has been an increasingly popular tool to monitor wildlife compared to the traditional methods such as satellite imagery-based sensing or GPS trackers. However, there still exist unsolved problems as to how the UAVs need to cover a spacious domain to detect animals as many as possible. In this paper, we propose the optimal transport-based wildlife monitoring strategy for a multi-UAV system, to prioritize monitoring areas while incorporating complementary information such as GPS trackers and satellite-based sensing. Through the proposed scheme, the UAVs can explore the large-size domain effectively and collaboratively with a given priority. The time-varying nature of wildlife due to their movements is modeled as a stochastic process, which is included in the proposed work to reflect the spatio-temporal evolution of their position estimation. In this way, the proposed monitoring plan can lead to wildlife monitoring with a high detection rate. Various simulation results including statistical data are provided to validate the proposed work. In all different simulations, it is shown that the proposed scheme significantly outperforms other UAV-based wildlife monitoring strategies in terms of the target detection rate up to 3.6 times.

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