[Paper Review] Intrinsic Wasserstein Correlation Analysis

Hang Zhou, Zhenhua Lin+1 more|arXiv (Cornell University)|May 31, 2021

Advanced Neuroimaging Techniques and Applications40 references4 citations

TL;DR

This paper introduces Intrinsic Wasserstein Correlation Analysis (IWCA), a novel framework for canonical correlation analysis of distribution-valued functional data using the intrinsic geometry of Wasserstein spaces. By extending tensor Hilbert space theory to non-linear Wasserstein spaces via Riemannian logarithmic maps and parallel transport, the method enables estimation of correlation and weight functions using FPCA and Tikhonov regularization, achieving minimax optimal convergence rates under regularity conditions.

ABSTRACT

We develop a framework of canonical correlation analysis for distribution-valued functional data within the geometry of Wasserstein spaces. Specifically, we formulate an intrinsic concept of correlation between random distributions, propose estimation methods based on functional principal component analysis (FPCA) and Tikhonov regularization, respectively, for the correlation and its corresponding weight functions, and establish the minimax convergence rates of the estimators. The key idea is to extend the framework of tensor Hilbert spaces to distribution-valued functional data to overcome the challenging issue raised by nonlinearity of Wasserstein spaces. The finite-sample performance of the proposed estimators is illustrated via simulation studies, and the practical merit is demonstrated via a study on the association of distributions of brain activities between two brain regions.

Motivation & Objective

• To develop a canonical correlation analysis framework for functional data that take values in Wasserstein spaces, which are non-linear and lack standard Hilbert space structure.
• To address the challenge of nonlinearity in Wasserstein spaces by leveraging their intrinsic Riemannian geometry, particularly through logarithmic maps and parallel transport.
• To propose estimators for the correlation and corresponding weight functions using functional principal component analysis (FPCA) and Tikhonov regularization.
• To establish minimax convergence rates for the proposed estimators under appropriate regularity conditions.
• To demonstrate the method's practical utility through simulation studies and an fMRI brain activity distribution analysis.

Proposed method

• Formulates an intrinsic concept of correlation between random distributions in Wasserstein space using the Riemannian structure compatible with optimal transport.
• Employs Riemannian logarithmic maps to linearize the Wasserstein space, enabling the use of tensor Hilbert space techniques in an intrinsic geometric setting.
• Applies functional principal component analysis (FPCA) to estimate the correlation structure and weight functions from observed distributional functional data.
• Uses Tikhonov regularization to stabilize the estimation of correlation and weight functions, particularly in high-dimensional or ill-conditioned settings.
• Extends the framework of tensor Hilbert spaces from finite-dimensional Riemannian manifolds to infinite-dimensional Wasserstein spaces via parallel transport and geodesic interpolation.
• Establishes theoretical convergence rates by analyzing eigenvalue decay and perturbation bounds under assumptions on eigenstructure and sample size.

Experimental results

Research questions

• RQ1 How can canonical correlation analysis be meaningfully defined for functional data that take values in a non-linear Wasserstein space?
• RQ2 What is the intrinsic geometric structure of correlation between random distributions in Wasserstein space, and how can it be estimated consistently?
• RQ3 Can FPCA and Tikhonov regularization be adapted to estimate correlation and weight functions in this non-linear setting with optimal convergence rates?
• RQ4 How does the nonlinearity of the Wasserstein space affect the theoretical properties of correlation estimators compared to classical Euclidean functional data analysis?
• RQ5 What is the minimax rate of convergence for correlation estimators in the Wasserstein functional data setting, and can it be achieved by the proposed method?

Key findings

• The proposed Intrinsic Wasserstein Correlation Analysis framework successfully extends canonical correlation analysis to distribution-valued functional data by leveraging the intrinsic Riemannian geometry of Wasserstein spaces.
• The method achieves minimax optimal convergence rates for both the correlation estimator and the corresponding weight functions under regularity conditions on eigenvalue decay and spectral gaps.
• Simulation studies confirm the finite-sample performance and robustness of the FPCA-based and Tikhonov-regularized estimators in estimating correlation and weight functions.
• The theoretical analysis establishes that the eigenvalue perturbation bounds and convergence rates are valid even in the infinite-dimensional setting, provided the eigenstructure satisfies polynomial decay conditions.
• The method is practically validated on fMRI data, demonstrating its ability to detect meaningful associations between the distributions of brain signal intensities in different brain regions.
• The paper corrects a gap in prior theoretical work on convergence rates by showing that the RKHS norm bound diverges under standard assumptions, highlighting the necessity of the proposed intrinsic geometric approach.