On the Wasserstein median of probability measures
- Computer Science
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
- Computer Science, MathematicsICML
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
- Computer ScienceJ. Mach. Learn. Res.
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
- Computer ScienceAAAI
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
- Mathematics, Computer ScienceArXiv
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
- Computer Science
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
- MathematicsJ. Multivar. Anal.
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
- Computer Science, Mathematics
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.
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
- Computer Science, Mathematics
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.
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
- Mathematics, Computer ScienceSIAM J. Math. Anal.
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,