[Paper Review] Construction of Non-asymptotic Confidence Sets in 2-Wasserstein Space

Johannes Ebert, Vladimir Spokoiny+1 more|arXiv (Cornell University)|Mar 10, 2017

Topological and Geometric Data Analysis21 references14 citations

TL;DR

This paper proposes a non-asymptotic bootstrap method for constructing confidence sets around empirical Wasserstein barycenters in 2-Wasserstein space, focusing on probability measures with commuting covariance matrices within a location-scatter family. The approach leverages multiplier bootstrap to enable valid inference and hypothesis testing, including change point detection, under finite-sample guarantees.

ABSTRACT

In this paper, we consider a probabilistic setting where the probability measures are considered to be random objects. We propose a procedure of construction non-asymptotic confidence sets for empirical barycenters in 2-Wasserstein space and develop the idea further to construction of a non-parametric two-sample test that is then applied to the detection of structural breaks in data with complex geometry. Both procedures mainly rely on the idea of multiplier bootstrap (Spokoiny and Zhilova (2015), Chernozhukov et al. (2014)). The main focus lies on probability measures that have commuting covariance matrices and belong to the same scatter-location family: we proof the validity of a bootstrap procedure that allows to compute confidence sets and critical values for a Wasserstein-based two-sample test.

Motivation & Objective

• To develop finite-sample confidence sets for empirical Wasserstein barycenters in 2-Wasserstein space, avoiding asymptotic approximations.
• To establish a valid bootstrap procedure for constructing critical values in nonparametric two-sample tests based on Wasserstein distance.
• To enable structural break detection in data with complex geometric structure using Wasserstein-based inference.
• To ensure theoretical validity of the bootstrap under the assumption of commuting covariance matrices within a location-scatter family.
• To provide a framework for inference on barycenters that is applicable in non-Euclidean, manifold-structured data settings such as shape analysis and image processing.

Proposed method

• Uses the multiplier bootstrap procedure as developed in Chernozhukov, Chetverikov, and Kato (2013) and Spokoiny and Zhilova (2015) for high-dimensional inference.
• Applies the bootstrap to the empirical Wasserstein barycenter to construct non-asymptotic confidence sets under the assumption of commuting covariance matrices.
• Relies on Gaussian approximation and anti-concentration inequalities to control the bootstrap error in high-dimensional settings.
• Employs matrix Bernstein inequality to bound the operator norm of empirical covariance matrix deviations.
• Derives a sub-exponential tail bound for the deviation of empirical quadratic forms, ensuring concentration under moment conditions.
• Constructs a two-sample test statistic based on the 2-Wasserstein distance between empirical barycenters, with critical values obtained via bootstrap.

Experimental results

Research questions

• RQ1 Can non-asymptotic confidence sets be constructed for empirical Wasserstein barycenters in 2-Wasserstein space under finite-sample conditions?
• RQ2 Is the multiplier bootstrap valid for inference on Wasserstein barycenters when the underlying probability measures have commuting covariance matrices?
• RQ3 Can the proposed bootstrap procedure be extended to detect structural breaks in data with complex geometric structure?
• RQ4 What are the finite-sample properties of the bootstrap-based two-sample test in the context of 2-Wasserstein barycenters?
• RQ5 How do the assumptions of commuting covariance matrices and the location-scatter family affect the validity and accuracy of the bootstrap inference?

Key findings

• The bootstrap procedure is proven valid for constructing confidence sets around Wasserstein barycenters under the condition that the underlying probability measures have commuting covariance matrices.
• The paper establishes a Kullback-Leibler divergence bound between two multivariate normal distributions with close covariance structures, showing that KL divergence is bounded by half the squared Frobenius norm of the difference in transformed covariance matrices.
• A matrix Bernstein inequality is applied to control the operator norm of the empirical covariance matrix deviation, yielding a high-probability bound of the form $\mathbb{P}(\|Z\|_{\text{op}} \geq \mathtt{C}\mathtt{x}/\sqrt{n}) \leq 2\exp\bigl(-(\mathtt{x}-6L\log(d))/6L\bigr{)}$.
• The sub-exponential tail bound ensures that deviations of empirical quadratic forms are well-controlled, with tail decay depending on the variance and sub-Gaussian tail parameter.
• Theoretical validity of the bootstrap is established under conditions on the eigenvalues of the ratio of covariance matrices, ensuring that $\|\Sigma_2^{-1/2}\Sigma_1\Sigma_2^{-1/2} - I_d\| \leq 1/2$ and $\operatorname{tr}((\Sigma_2^{-1/2}\Sigma_1\Sigma_2^{-1/2} - I_d)^2) \leq \rho_\Sigma^2/2$.
• The method enables non-asymptotic inference in high-dimensional settings with compactly supported random vectors, as demonstrated through simulation and real data experiments on coverage probability and change point detection.