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

arXiv: https://arxiv.org/abs/2607.28981 · HTML: https://mlsanity.com/paper/2607.28981/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 13 of its 43 references that are in the map.
- First posted on arXiv 31 Jul 2026.

## Builds on (most cited first, 13 of 13)

- [Kernel Methods on the Riemannian Manifold of Symmetric Positive Definite Matrices](https://mlsanity.com/paper/1412.4172/) (2014)
- [Nonlinear Sufficient Dimension Reduction for Distribution-on-Distribution Regression](https://mlsanity.com/paper/2207.04613/) (2022)
- [Nonlinear global Fr'echet regression for random objects via weak conditional expectation](https://mlsanity.com/paper/2310.07817/) (2023)
- [Characteristic kernels on Hilbert spaces, Banach spaces, and on sets of measures](https://mlsanity.com/paper/2206.07588/) (2022)
- [Distance-based and RKHS-based Dependence Metrics in High Dimension](https://mlsanity.com/paper/1902.03291/) (2019)
- [Doubly robust estimation of causal effects for random object outcomes with continuous treatments](https://mlsanity.com/paper/2506.22754/) (2025)
- [Metric Statistics: Exploration and Inference for Random Objects With Distance Profiles](https://mlsanity.com/paper/2202.06117/) (2022)
- [Wasserstein F-tests for Fr'echet regression on Bures-Wasserstein manifolds](https://mlsanity.com/paper/2404.03878/) (2024)
- [Nonlinear function-on-function regression by RKHS](https://mlsanity.com/paper/2207.08211/) (2022)
- [Association and Independence Test for Random Objects](https://mlsanity.com/paper/2505.01983/) (2025)
- [Nonparametric Statistical Inference via Metric Distribution Function in Metric Spaces](https://mlsanity.com/paper/2107.07317/) (2021)
- [A Copula Graphical Model for Multi-Attribute Data using Optimal Transport](https://mlsanity.com/paper/2404.06735/) (2024)
- [High-dimensional log contrast models with measurement errors](https://mlsanity.com/paper/2407.15084/) (2024)

## Abstract (arXiv)

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.