Hilbert space embeddings of independence tests and interaction measures of several variables

2024/11/13 by Jean Carlo Guella, Guella, Jean Carlo
Computer Science · Mathematics · #30L05 #43A35 #44A10 #46H20 #FOS: Mathematics #Functional Analysis (math.FA) #Fuzzy Systems and Optimization #Matrix Theory and Algorithms #Probability (math.PR) #advanced mathematical theories

paper · pdf · doi:10.48550/arxiv.2411.08653

openalex publication_date 2024/11/13 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We present a unified theoretical framework for kernel-based measures of dependence on product spaces. Building on the ideas underlying distance covariance, distance multivariance, and the Hilbert-Schmidt Independence Criterion (HSIC), we define a new family of kernels on an n-fold Cartesian product, termed positive definite independent of order k (PDI_k kernels). These kernels extend the concepts of positive definite and conditionally negative definite kernels to higher orders and provide the foundation for generalized independence and interaction tests, such as the generalized Lancaster interaction of order k (Λ_kⁿ), and the Streitberg interaction (Σ). Our analysis focuses on the continuous setting, where we prove a Kernel Mean Embedding Theorem for PDI_k kernels and establish the corresponding integrability restrictions. Based on these results, we characterize how the Kronecker products of PDI kernels behave.

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