scholarly journals Squared quadratic Wasserstein distance: optimal couplings and Lions differentiability

2020 ◽  
Vol 24 ◽  
pp. 703-717
Author(s):  
Aurélien Alfonsi ◽  
Benjamin Jourdain
Keyword(s):  
Optimal Transport ◽  
Direct Proof ◽  
Invariant Function ◽  
Wasserstein Distance ◽  
Collège De France ◽  
Fréchet Differential ◽  
Probabilistic Proof ◽  
Optimal Coupling ◽  
Optimal Transport Map ◽  
Appl Math

In this paper, we remark that any optimal coupling for the quadratic Wasserstein distance W22(μ,ν) between two probability measures μ and ν with finite second order moments on ℝd is the composition of a martingale coupling with an optimal transport map 𝛵. We check the existence of an optimal coupling in which this map gives the unique optimal coupling between μ and 𝛵#μ. Next, we give a direct proof that σ ↦ W22(σ,ν) is differentiable at μ in the Lions (Cours au Collège de France. 2008) sense iff there is a unique optimal coupling between μ and ν and this coupling is given by a map. It was known combining results by Ambrosio, Gigli and Savaré (Lectures in Mathematics ETH Zürich. Birkhäuser Verlag, Basel, 2005) and Ambrosio and Gangbo (Comm. Pure Appl. Math., 61:18–53, 2008) that, under the latter condition, geometric differentiability holds. Moreover, the two notions of differentiability are equivalent according to the recent paper of Gangbo and Tudorascu (J. Math. Pures Appl. 125:119–174, 2019). Besides, we give a self-contained probabilistic proof that mere Fréchet differentiability of a law invariant function F on L2(Ω, ℙ; ℝd) is enough for the Fréchet differential at X to be a measurable function of X.

scholarly journals Entropy-Regularized Optimal Transport on Multivariate Normal and q-normal Distributions

Entropy ◽  
2021 ◽  
Vol 23 (3) ◽  
pp. 302
Author(s):  
Qijun Tong ◽  
Kei Kobayashi
Keyword(s):  
Explicit Form ◽  
Optimal Transport ◽  
General Setting ◽  
Theoretical Aspect ◽  
Wasserstein Distance ◽  
Transport Cost ◽  
Multivariate Normal ◽  
Statistical Efficiency ◽  
Normal Distributions ◽  
Optimal Coupling

The distance and divergence of the probability measures play a central role in statistics, machine learning, and many other related fields. The Wasserstein distance has received much attention in recent years because of its distinctions from other distances or divergences. Although computing the Wasserstein distance is costly, entropy-regularized optimal transport was proposed to computationally efficiently approximate the Wasserstein distance. The purpose of this study is to understand the theoretical aspect of entropy-regularized optimal transport. In this paper, we focus on entropy-regularized optimal transport on multivariate normal distributions and q-normal distributions. We obtain the explicit form of the entropy-regularized optimal transport cost on multivariate normal and q-normal distributions; this provides a perspective to understand the effect of entropy regularization, which was previously known only experimentally. Furthermore, we obtain the entropy-regularized Kantorovich estimator for the probability measure that satisfies certain conditions. We also demonstrate how the Wasserstein distance, optimal coupling, geometric structure, and statistical efficiency are affected by entropy regularization in some experiments. In particular, our results about the explicit form of the optimal coupling of the Tsallis entropy-regularized optimal transport on multivariate q-normal distributions and the entropy-regularized Kantorovich estimator are novel and will become the first step towards the understanding of a more general setting.

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 Pollution Transfer as Optimal Mass Transport Problem

2016 ◽  
Vol 8 (6) ◽  
pp. 58
Author(s):  
L. Ndiaye ◽  
Mb. Ndiaye ◽  
A. Sy ◽  
D. Seck
Keyword(s):  
Porous Media ◽  
Optimal Transport ◽  
Existence Of Solution ◽  
Nonlinear Pde ◽  
Mathematical Framework ◽  
Sufficient Condition ◽  
Mass Transportation ◽  
Optimal Mass Transport ◽  
Transportation Theory ◽  
Optimal Transport Map

In this paper, we use mass transportation theory to study pollution  transfer in  porous media.  We show   the existence of a $L^2-$regular vector field defined by a $W^{1, 1}-$ optimal transport map. A sufficient condition for solvability of our model, is given by   a (non homogeneous) transport equation with  a  source defined by a measure. The mathematical framework used, allows us to  show in some specifical cases, existence of solution for  a nonlinear PDE deriving from the modelling. And we end by numerical simulations.

scholarly journals Gromov-Wasserstein optimal transport to align single-cell multi-omics data

2020 ◽  
Author(s):  
Pinar Demetci ◽  
Rebecca Santorella ◽  
Björn Sandstede ◽  
William Stafford Noble ◽  
Ritambhara Singh
Keyword(s):  
Single Cell ◽  
Optimal Transport ◽  
Learning Algorithm ◽  
State Of The Art ◽  
Single Cells ◽  
Wasserstein Distance ◽  
Cell Alignment ◽  
Shared Space ◽  
Real World Datasets ◽  
Unsupervised Algorithms

AbstractData integration of single-cell measurements is critical for understanding cell development and disease, but the lack of correspondence between different types of measurements makes such efforts challenging. Several unsupervised algorithms can align heterogeneous single-cell measurements in a shared space, enabling the creation of mappings between single cells in different data domains. However, these algorithms require hyperparameter tuning for high-quality alignments, which is difficult in an unsupervised setting without correspondence information for validation. We present Single-Cell alignment using Optimal Transport (SCOT), an unsupervised learning algorithm that uses Gromov Wasserstein-based optimal transport to align single-cell multi-omics datasets. We compare the alignment performance of SCOT with state-of-the-art algorithms on four simulated and two real-world datasets. SCOT performs on par with state-of-the-art methods but is faster and requires tuning fewer hyperparameters. Furthermore, we provide an algorithm for SCOT to use Gromov Wasserstein distance to guide the parameter selection. Thus, unlike previous methods, SCOT aligns well without using any orthogonal correspondence information to pick the hyperparameters. Our source code and scripts for replicating the results are available at https://github.com/rsinghlab/SCOT.

scholarly journals Optimal Transport with Dimensionality Reduction for Domain Adaptation

Symmetry ◽  
2020 ◽  
Vol 12 (12) ◽  
pp. 1994
Author(s):  
Ping Li ◽  
Zhiwei Ni ◽  
Xuhui Zhu ◽  
Juan Song ◽  
Wenying Wu
Keyword(s):  
Dimensionality Reduction ◽  
Optimal Transport ◽  
Domain Adaptation ◽  
Wasserstein Distance ◽  
Local Information ◽  
Target Domain ◽  
Source Domain ◽  
Second Stage ◽  
Cross Domain ◽  
Feature Based

Domain adaptation manages to learn a robust classifier for target domain, using the source domain, but they often follow different distributions. To bridge distribution shift between the two domains, most of previous works aim to align their feature distributions through feature transformation, of which optimal transport for domain adaptation has attract researchers’ interest, as it can exploit the local information of the two domains in the process of mapping the source instances to the target ones by minimizing Wasserstein distance between their feature distributions. However, it may weaken the feature discriminability of source domain, thus degrade domain adaptation performance. To address this problem, this paper proposes a two-stage feature-based adaptation approach, referred to as optimal transport with dimensionality reduction (OTDR). In the first stage, we apply the dimensionality reduction with intradomain variant maximization but source intraclass compactness minimization, to separate data samples as much as possible and enhance the feature discriminability of the source domain. In the second stage, we leverage optimal transport-based technique to preserve the local information of the two domains. Notably, the desirable properties in the first stage can mitigate the degradation of feature discriminability of the source domain in the second stage. Extensive experiments on several cross-domain image datasets validate that OTDR is superior to its competitors in classification accuracy.

scholarly journals A non-exponential extension of Sanov’s theorem via convex duality

2020 ◽  
Vol 52 (1) ◽  
pp. 61-101
Author(s):  
Daniel Lacker
Keyword(s):  
Optimal Transport ◽  
Large Deviation ◽  
Optimization Problems ◽  
Risk Measures ◽  
Approximate Solutions ◽  
Variational Problems ◽  
Dual Pair ◽  
Wasserstein Distance ◽  
Empirical Measures ◽  
Empirical Distributions

AbstractThis work is devoted to a vast extension of Sanov’s theorem, in Laplace principle form, based on alternatives to the classical convex dual pair of relative entropy and cumulant generating functional. The abstract results give rise to a number of probabilistic limit theorems and asymptotics. For instance, widely applicable non-exponential large deviation upper bounds are derived for empirical distributions and averages of independent and identically distributed samples under minimal integrability assumptions, notably accommodating heavy-tailed distributions. Other interesting manifestations of the abstract results include new results on the rate of convergence of empirical measures in Wasserstein distance, uniform large deviation bounds, and variational problems involving optimal transport costs, as well as an application to error estimates for approximate solutions of stochastic optimization problems. The proofs build on the Dupuis–Ellis weak convergence approach to large deviations as well as the duality theory for convex risk measures.

scholarly journals An unbalanced optimal transport splitting scheme for general advection-reaction-diffusion problems

2019 ◽  
Vol 25 ◽  
pp. 8 ◽  
Author(s):  
Thomas Gallouët ◽  
Maxime Laborde ◽  
Léonard Monsaingeon
Keyword(s):  
Optimal Transport ◽  
Diffusion Processes ◽  
Reaction Diffusion ◽  
Wasserstein Distance ◽  
Diffusion Problem ◽  
Constructive Method ◽  
Interacting Species ◽  
General Scalar ◽  
Minimizing Movements ◽  
Rao Distance

In this paper, we show that unbalanced optimal transport provides a convenient framework to handle reaction and diffusion processes in a unified metric setting. We use a constructive method, alternating minimizing movements for the Wasserstein distance and for the Fisher-Rao distance, and prove existence of weak solutions for general scalar reaction-diffusion-advection equations. We extend the approach to systems of multiple interacting species, and also consider an application to a very degenerate diffusion problem involving a Gamma-limit. Moreover, some numerical simulations are included.

scholarly journals Transfer Learning for Wireless Fingerprinting Localization Based on Optimal Transport

Sensors ◽  
2020 ◽  
Vol 20 (23) ◽  
pp. 6994
Author(s):  
Siqi Bai ◽  
Yongjie Luo ◽  
Qun Wan
Keyword(s):  
Transfer Learning ◽  
Optimal Transport ◽  
Wasserstein Distance ◽  
Target Domain ◽  
The Public ◽  
Empirical Distributions ◽  
Feature Based ◽  
And Performance ◽  
The Cost ◽  
Radio Map

Wireless fingerprinting localization (FL) systems identify locations by building radio fingerprint maps, aiming to provide satisfactory location solutions for the complex environment. However, the radio map is easy to change, and the cost of building a new one is high. One research focus is to transfer knowledge from the old radio maps to a new one. Feature-based transfer learning methods help by mapping the source fingerprint and the target fingerprint to a common hidden domain, then minimize the maximum mean difference (MMD) distance between the empirical distributions in the latent domain. In this paper, the optimal transport (OT)-based transfer learning is adopted to directly map the fingerprint from the source domain to the target domain by minimizing the Wasserstein distance so that the data distribution of the two domains can be better matched and the positioning performance in the target domain is improved. Two channel-models are used to simulate the transfer scenarios, and the public measured data test further verifies that the transfer learning based on OT has better accuracy and performance when the radio map changes in FL, indicating the importance of the method in this field.

scholarly journals Manifold-Valued Image Generation with Wasserstein Generative Adversarial Nets

2019 ◽  
Vol 33 ◽  
pp. 3886-3893 ◽  
Author(s):  
Zhiwu Huang ◽  
Jiqing Wu ◽  
Luc Van Gool
Keyword(s):  
Optimal Transport ◽  
Diffusion Tensor ◽  
Color Image ◽  
Wasserstein Distance ◽  
Generative Models ◽  
Color Images ◽  
Image Generation ◽  
Generative Modeling ◽  
Benchmark Datasets ◽  
Complete Manifolds

Generative modeling over natural images is one of the most fundamental machine learning problems. However, few modern generative models, including Wasserstein Generative Adversarial Nets (WGANs), are studied on manifold-valued images that are frequently encountered in real-world applications. To fill the gap, this paper first formulates the problem of generating manifold-valued images and exploits three typical instances: hue-saturation-value (HSV) color image generation, chromaticity-brightness (CB) color image generation, and diffusion-tensor (DT) image generation. For the proposed generative modeling problem, we then introduce a theorem of optimal transport to derive a new Wasserstein distance of data distributions on complete manifolds, enabling us to achieve a tractable objective under the WGAN framework. In addition, we recommend three benchmark datasets that are CIFAR-10 HSV/CB color images, ImageNet HSV/CB color images, UCL DT image datasets. On the three datasets, we experimentally demonstrate the proposed manifold-aware WGAN model can generate more plausible manifold-valued images than its competitors.

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