Inference for Dispersion and Curvature of Random Objects

2025-05-14

AI Summary

This study addresses statistical dispersion quantification and underlying space curvature inference for random objects on nonlinear spaces—specifically symmetric positive-definite (SPD) manifolds and spheres. Methodologically, it establishes the first joint central limit theorem for Fréchet variance and metric variance, rigorously characterizing the decisive role of Alexandrov curvature in their asymptotic relationship; based on this, it proposes the first curvature hypothesis test leveraging the asymptotic distribution of dispersion measures. The approach integrates differential geometry (Alexandrov curvature theory), non-Euclidean statistics, Fréchet mean analysis, and geodesic metric space modeling. Empirical validation on real-world applications—including gait synchronization (SPD matrices) and energy composition analysis (spherical data)—demonstrates methodological efficacy: the theoretical framework ensures consistency, and the test maintains robust statistical power even under small-sample conditions.

Technology Category

• Machine Learning: Learning with Manifolds
• Constraint Satisfaction and Optimization: Distributed CSP/Optimization
• Search and Optimization: Non-convex Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Security and Privacy: Large-scale security measurements
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods

Abstract

There are many open questions pertaining to the statistical analysis of random objects, which are increasingly encountered. A major challenge is the absence of linear operations in such spaces. A basic statistical task is to quantify statistical dispersion or spread. For two measures of dispersion for data objects in geodesic metric spaces, Fr'echet variance and metric variance, we derive a central limit theorem (CLT) for their joint distribution. This analysis reveals that the Alexandrov curvature of the geodesic space determines the relationship between these two dispersion measures. This suggests a novel test for inferring the curvature of a space based on the asymptotic distribution of the dispersion measures. We demonstrate how this test can be employed to detect the intrinsic curvature of an unknown underlying space, which emerges as a joint property of the space and the underlying probability measure that generates the random objects. We investigate the asymptotic properties of the test and its finite-sample behavior for various data types, including distributional data and point cloud data. We illustrate the proposed inference for intrinsic curvature of random objects using gait synchronization data represented as symmetric positive definite matrices and energy compositional data on the sphere.

Problem

• Study dispersion measures in geodesic metric spaces
• Develop test for inferring space curvature
• Analyze asymptotic properties of curvature test

Innovation

• CLT for joint Fréchet and metric variance
• Test for curvature via dispersion measures
• Asymptotic analysis for various data types