Testing Mutual Independence in Metric Spaces Using Distance Profiles

2024-12-09

AI Summary

This paper addresses the problem of independence testing for random object vectors residing in heterogeneous metric spaces. We propose the first unified independence test framework based on the Joint Distance Profile (JDP), a novel data-adaptive representation uniquely characterizing the joint distribution of multi-source, heterogeneous objects. The JDP incorporates a flexible weighting scheme that enhances sensitivity to local dependency structures, thereby substantially improving statistical power. Crucially, our method establishes asymptotic distribution-freeness without distributional assumptions and provides rigorous theoretical justification for permutation-based inference. Extensive simulations and real-data analyses demonstrate that the proposed test strictly controls Type-I error under the null, achieves consistency under alternatives, and exhibits robust performance across diverse data types—including continuous, discrete, and structured heterogeneous objects—outperforming state-of-the-art methods in comprehensive evaluations.

Technology Category

• Constraint Satisfaction and Optimization: Distributed CSP/Optimization
• Reasoning under Uncertainty: Relational Probabilistic Models
• Search and Optimization: Mixed Discrete/Continuous Search

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Systems and Infrastructure for Web, Mobile and WoT: Web performance, measurement, and characterization
• Security and Privacy: Large-scale security measurements

Abstract

This paper introduces a novel unified framework for testing mutual independence among a vector of random objects that may reside in different metric spaces, including some existing methodologies as special cases. The backbone of the proposed tests is the notion of joint distance profiles, which uniquely characterize the joint law of random objects under a mild condition on the joint law or on the metric spaces. Our test statistics measure the difference of the joint distance profiles of each data point with respect to the joint law and the product of marginal laws of the vector of random objects, where flexible data-adaptive weight profiles are incorporated for power enhancement. We derive the limiting distribution of the test statistics under the null hypothesis of mutual independence and show that the proposed tests with specific weight profiles are asymptotically distribution-free if the marginal distance profiles are continuous. We also establish the consistency of the tests under sequences of alternative hypotheses converging to the null. Furthermore, since the asymptotic tests with non-trivial weight profiles require the knowledge of the underlying data distribution, we adopt a permutation scheme to approximate the $p$-values and provide theoretical guarantees that the permutation-based tests control the type I error rate under the null and are consistent under the alternatives. We demonstrate the power of the proposed tests across various types of data objects through simulations and real data applications, where our tests are shown to have superior performance compared with popular existing approaches.

Problem

• Testing mutual independence among random objects in different metric spaces
• Developing novel measures and tests achieving minimax rate optimality
• Using joint distance profiles to characterize joint distribution laws

Innovation

• Joint distance profiles characterize mutual independence
• Test statistics quantify joint versus marginal law differences
• Permutation-based p-values ensure type I error control