# Kernel Partial Correlation Coefficient -- a Measure of Conditional Dependence

arXiv: https://arxiv.org/abs/2012.14804 · HTML: https://mlsanity.com/paper/2012.14804/

## Key facts

- Cited by 21 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 29 Dec 2020.

## Built on by (most cited first, 21 of 21)

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- [Causal Domain Adaptation with Copula Entropy based Conditional Independence Test](https://mlsanity.com/paper/2202.13482/) (2022)
- [Evaluating Independence and Conditional Independence Measures](https://mlsanity.com/paper/2205.07253/) (2022)
- [Randomized Quasi-Monte Carlo Features for Kernel Approximation](https://mlsanity.com/paper/2503.06041/) (2025)
- [One-shot Conditional Sampling: MMD meets Nearest Neighbors](https://mlsanity.com/paper/2509.25507/) (2025)
- [Kernel Integrated R2: A Measure of Dependence](https://mlsanity.com/paper/2602.22985/) (2026)
- [Causality-Encoded Diffusion Models for Interventional Sampling and Edge Inference](https://mlsanity.com/paper/2604.21843/) (2026)
- [Detecting Changes in Causal Dependence with Kernels and Copulas](https://mlsanity.com/paper/2605.05809/) (2026)

## Abstract (arXiv)

In this paper we propose and study a class of simple, nonparametric, yet interpretable measures of conditional dependence between two random variables Y and Z given a third variable X, all taking values in general topological spaces. The population version of any of these measures captures the strength of conditional dependence and it is 0 if and only if Y and Z are conditionally independent given X, and 1 if and only if Y is a measurable function of Z and X. Thus, our measure -- which we call kernel partial correlation (KPC) coefficient -- can be thought of as a nonparametric generalization of the partial correlation coefficient that possesses the above properties when (X,Y,Z) is jointly normal. We describe two consistent methods of estimating KPC. Our first method utilizes the general framework of geometric graphs, including K-nearest neighbor graphs and minimum spanning trees. A sub-class of these estimators can be computed in near linear time and converges at a rate that automatically adapts to the intrinsic dimension(s) of the underlying distribution(s). Our second strategy involves direct estimation of conditional mean embeddings using cross-covariance operators in the reproducing kernel Hilbert spaces. Using these empirical measures we develop forward stepwise (high-dimensional) nonlinear variable selection algorithms. We show that our algorithm, using the graph-based estimator, yields a provably consistent model-free variable selection procedure, even in the high-dimensional regime when the number of covariates grows exponentially with the sample size, under suitable sparsity assumptions. Extensive simulation and real-data examples illustrate the superior performance of our methods compared to existing procedures. The recent conditional dependence measure proposed by Azadkia and Chatterjee (2019) can be viewed as a special case of our general framework.