Distance Profile Embedding for Independence and Conditional Independence Testing of Random Objects

• URL: http://arxiv.org/abs/2607.28981v1
• Date: Fri, 31 Jul 2026 03:12:47 GMT
• Title: Distance Profile Embedding for Independence and Conditional Independence Testing of Random Objects
• Abstract summary: Testing independence or conditional independence is fundamental to statistical inference, yet existing methods face a difficult trade-off between geometric flexibility and theoretical tractability.<n>We introduce the Distance Profile Embedding (DPE), a novel representation that maps random objects from general metric spaces into a space of square-integrable functions.<n>We develop a unified framework for marginal and conditional independence testing of random objects that enjoys a rigorous theory for both size and power.
• Score: 8.679002268074683
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: Testing independence or conditional independence is fundamental to statistical inference, yet existing methods for non-Euclidean random objects often face a difficult trade-off between geometric flexibility and theoretical tractability. We introduce the Distance Profile Embedding (DPE), a novel representation that maps random objects from general metric spaces into a Hilbert space of square-integrable functions. We prove that this mapping is injective and preserves full distributional information without requiring isometric Hilbert embeddings or one-to-one correspondence conditions. Leveraging the DPE, we develop a unified framework for marginal and conditional independence testing of random objects that enjoys a rigorous asymptotic theory for both size and power. Notably, our framework is the first in the literature to accommodate object-valued conditioning variables when testing conditional independence, overcoming the Euclidean or Hilbertian constraints of existing methodologies. We facilitate the calculation of analytic $p$-values using closed-form asymptotic null distributions, which avoids the computational burden of permutation tests common in existing metric-based methods. The numerical properties of our methods are demonstrated through both simulations and two real-world applications involving gut microbiome compositions and global human mortality distributions, respectively.

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