A Geometric study of Wasserstein spaces: Euclidean spaces

Kloeckner, Benoît

doi:10.2422/2036-2145.2010.2.03

Cited by 40 publications

(73 citation statements)

References 19 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…In case of Riemannian manifolds the existence, uniqueness, and the convexity of the center of mass is guaranteed (Afsari, 2011;Pennec et al, 2018). In a space with a negative or 0 curvature, or in a Hadamard space the existence and uniqueness of the Fréchet means are also shown (Bhattacharya et al, 2003;Bhattacharya and Patrangenaru, 2005;Patrangenaru and Ellingson, 2015;Sturm, 2003;Kloeckner, 2010).…”

Section: Assumptions (A1)-(a2) Are Necessary To Show That the Interme...mentioning

confidence: 93%

Concurrent Object Regression

Bhattacharjee¹,

Mueller²

2021

Preprint

View full text Add to dashboard Cite

Modern-day problems in statistics often face the challenge of exploring and analyzing complex non-Euclidean object data that do not conform to vector space structures or operations. Examples of such data objects include covariance matrices, graph Laplacians of networks and univariate probability distribution functions. In the current contribution a new concurrent regression model is proposed to characterize the time-varying relation between an object in a general metric space (as response) and a vector in R p (as predictor), where concepts from Fréchet regression is employed. Concurrent regression has been a well-developed area of research for Euclidean predictors and responses, with many important applications for longitudinal studies and functional data. We develop generalized versions of both global least squares regression and locally weighted least squares smoothing in the context of concurrent regression for responses which are situated in general metric spaces and propose estimators that can accommodate sparse and/or irregular designs. Consistency results are demonstrated for sample estimates of appropriate population targets along with the corresponding rates of convergence. The proposed models are illustrated with mortality data and resting state functional Magnetic Resonance Imaging data (fMRI) as responses.

show abstract

Section: Assumptions (A1)-(a2) Are Necessary To Show That the Interme...mentioning

confidence: 93%

Concurrent Object Regression

Bhattacharjee¹,

Mueller²

2021

Preprint

View full text Add to dashboard Cite

show abstract

“…Similarly, the additional term 16) does not show up in Theorem 3 because roughly speaking, the space W 2 (I) has zero curvature (Kloeckner, 2010).…”

Section: Identification and Estimationmentioning

confidence: 98%

Causal Inference on Distribution Functions

Lin¹,

Kong²,

Wang³

2021

Preprint

View full text Add to dashboard Cite

Understanding causal relationships is one of the most important goals of modern science.So far, the causal inference literature has focused almost exclusively on outcomes coming from a linear space, most commonly the Euclidean space. However, it is increasingly common that complex datasets collected through electronic sources, such as wearable devices and medical imaging, cannot be represented as data points from linear spaces.In this paper, we present a formal definition of causal effects for outcomes from nonlinear spaces, with a focus on the Wasserstein space of cumulative distribution functions.We develop doubly robust estimators and associated asymptotic theory for these causal effects. Our framework extends to outcomes from certain Riemannian manifolds. As an illustration, we use our framework to quantify the causal effect of marriage on physical activity patterns using wearable device data collected through the National Health and Nutrition Examination Survey.

show abstract

“…What do (scaled) isometries look like in the 2-Wasserstein space W 2 (R d )? This has been answered in Kloeckner (2010). Let us give a brief overview of these results.…”

Section: Identificationmentioning

confidence: 98%

“…The main assumption is that the production functions mapping the unobservables to the observables are scaled isometries in Wasserstein space. Kloeckner (2010) shows that isometries in Wasserstein space over higher-dimensional Euclidean space are similar to isometries in Euclidean spaces, i.e. maps composed of rotations, shifts, and reflections (Novikov & Taimanov 2006, chapter 1).…”

Section: Introductionmentioning

confidence: 99%

Distributional synthetic controls

Gunsilius¹

2020

Preprint

View full text Add to dashboard Cite

This article extends the method of synthetic controls to probability measures. The distribution of the synthetic control group is obtained as the optimally weighted barycenter in Wasserstein space of the distributions of the control groups which minimizes the distance to the distribution of the treatment group. It can be applied to settings with disaggregated-or aggregated (functional) data. The method produces a generically unique counterfactual distribution when the data are continuously distributed. A basic representation of the barycenter provides a computationally efficient implementation via a straightforward tensor-variate regression approach. In addition, identification results are provided that also shed new light on the classical synthetic controls estimator. As an illustration the approach estimates the aggregate effect of the legalization of cannabis on the distribution of household income in Colorado one year after Amendment 64.

show abstract

A Geometric study of Wasserstein spaces: Euclidean spaces

Cited by 40 publications

References 19 publications

Concurrent Object Regression

Concurrent Object Regression

Causal Inference on Distribution Functions

Distributional synthetic controls

Contact Info

Product

Resources

About