An Invitation to Statistics in Wasserstein Space

@inproceedings{Panaretos2020AnIT,
  title={An Invitation to Statistics in Wasserstein Space},
  author={Victor M. Panaretos and Yoav Zemel},
  year={2020}
}

On the Wasserstein median of probability measures

TLDR
This work establishes existence and consistency of the Wasserstein median, an equivalent of Fr´echet median under the 2-Wasserstein metric, and proposes a generic algorithm that makes use of any established routine for the Wassadstein barycenter in an iterative manner and proves its convergence.

Measure Estimation in the Barycentric Coding Model

TLDR
This paper provides novel geometrical, statistical, and computational insights for measure estimation under the barycentric coding model (BCM), and establishes an algorithm for solving for the coordinates in the BCM when all the measures are observed empirically via i.i.d. samples.

Projected Statistical Methods for Distributional Data on the Real Line with the Wasserstein Metric

We present a novel class of projected methods, to perform statistical analysis on a data set of probability distributions on the real line, with the 2-Wasserstein metric. We focus in particular on

Fast PCA in 1-D Wasserstein Spaces via B-splines Representation and Metric Projection

TLDR
A novel definition of Principal Component Analysis in the Wasserstein space is proposed that yields a straightforward optimization problem that is extremely fast to compute and performs similarly to the ones already proposed in the literature while retaining a much smaller computational cost.

Graphical and uniform consistency of estimated optimal transport plans

A general theory is provided delivering convergence of maximal cyclically monotone mappings containing the supports of coupling measures of sequences of pairs of possibly random probability measures

Wasserstein multivariate auto-regressive models for modeling distributional time series and its application in graph learning

TLDR
A new auto-regressive model for the statistical analysis of multivariate distributional time series using the theory of iterated random function systems and a consistent estimator for the model coefficient is proposed.

Nonlinear Sufficient Dimension Reduction for Distribution-on-Distribution Regression

TLDR
A novel framework for nonlinear sufficient dimension reduction where both the predictor and the response are distributional data, which are modeled as members of a metric space to build universal kernels on the metric spaces.

Measuring dependence between random vectors via optimal transport

On some connections between Esscher's tilting, saddlepoint approximations, and optimal transportation: a statistical perspective

We showcase some unexplored connections between saddlepoint approximations, measure transportation, and some key topics in information theory. To bridge these different areas, we review selectively

Kantorovich-Rubinstein distance and barycenter for finitely supported measures: Foundations and Algorithms

TLDR
A systematic discussion of a generalized barycenter based on a variant of unbalanced optimal transport (UOT) that allows for mass creation and destruction modeled by some cost parameter and its structure to be explicitly specified by the support of the input measures.
...

References

SHOWING 1-10 OF 10 REFERENCES

A geometrical approach to monotone functions in R n

This paper is concerned with the fine properties of monotone functions on R. We study the continuity and differentiability properties of these functions, the approximability properties, the structure

Optimal Transport: Old and New

Couplings and changes of variables.- Three examples of coupling techniques.- The founding fathers of optimal transport.- Qualitative description of optimal transport.- Basic properties.- Cyclical

Gradient Flows: In Metric Spaces and in the Space of Probability Measures

Notation.- Notation.- Gradient Flow in Metric Spaces.- Curves and Gradients in Metric Spaces.- Existence of Curves of Maximal Slope and their Variational Approximation.- Proofs of the Convergence

Real Analysis and Probability

1. Foundations: set theory 2. General topology 3. Measures 4. Integration 5. Lp spaces: introduction to functional analysis 6. Convex sets and duality of normed spaces 7. Measure, topology, and

Amplitude and phase variation of point processes

TLDR
A key element in this approach is to demonstrate that when the classical phase variation assumptions of Functional Data Analysis are applied to the point process case, they become equivalent to conditions interpretable through the prism of the theory of optimal transportation of measure.

Point Processes and Their Statistical Inference

  • A. Karr
  • Mathematics, Computer Science
  • 1988

Probability: Theory and Examples

This book is an introduction to probability theory covering laws of large numbers, central limit theorems, random walks, martingales, Markov chains, ergodic theorems, and Brownian motion. It is a

Barycenters in the Wasserstein Space

TLDR
This paper provides existence, uniqueness, characterizations, and regularity of the barycenter and relates it to the multimarginal optimal transport problem considered by Gangbo and Świech in [Comm. Pure Appl. Math., 51 (1998), pp. 23–45].

Real Analysis and Probability, volume 74

  • Cambridge University Press,
  • 2002