# Kernel Integrated R2: A Measure of Dependence

arXiv: https://arxiv.org/abs/2602.22985 · HTML: https://mlsanity.com/paper/2602.22985/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 28 of its 112 references that are in the map.
- First posted on arXiv 26 Feb 2026.

## Builds on (most cited first, 28 of 28)

- [A Measure-Theoretic Approach to Kernel Conditional Mean Embeddings](https://mlsanity.com/paper/2002.03689/) (2020)
- [Kernels for sequentially ordered data](https://mlsanity.com/paper/1601.08169/) (2016)
- [Graph Kernels: State-of-the-Art and Future Challenges](https://mlsanity.com/paper/2011.03854/) (2020)
- [Characteristic and Universal Tensor Product Kernels](https://mlsanity.com/paper/1708.08157/) (2017)
- [Kernel-based Tests for Joint Independence](https://mlsanity.com/paper/1603.00285/) (2016)
- [A simple measure of conditional dependence](https://mlsanity.com/paper/1910.12327/) (2019)
- [Large-Scale Kernel Methods for Independence Testing](https://mlsanity.com/paper/1606.07892/) (2016)
- [A Rigorous Theory of Conditional Mean Embeddings](https://mlsanity.com/paper/1912.00671/) (2019)
- [The Randomized Dependence Coefficient](https://mlsanity.com/paper/1304.7717/) (2013)
- [Multivariate Rank-based Distribution-free Nonparametric Testing using Measure Transportation](https://mlsanity.com/paper/1909.08733/) (2019)
- [Kernel Partial Correlation Coefficient -- a Measure of Conditional Dependence](https://mlsanity.com/paper/2012.14804/) (2020)
- [On boosting the power of Chatterjee's rank correlation](https://mlsanity.com/paper/2108.06828/) (2021)
- [BET on Independence](https://mlsanity.com/paper/1610.05246/) (2016)
- [Kernelized Cumulants: Beyond Kernel Mean Embeddings](https://mlsanity.com/paper/2301.12466/) (2023)
- [Symmetric Rank Covariances: a Generalised Framework for Nonparametric Measures of Dependence](https://mlsanity.com/paper/1708.05653/) (2017)
- [On Testing Independence and Goodness-of-fit in Linear Models](https://mlsanity.com/paper/1302.5831/) (2013)
- [Generalized R-squared for Detecting Dependence](https://mlsanity.com/paper/1604.02736/) (2016)
- [A survey of some recent developments in measures of association](https://mlsanity.com/paper/2211.04702/) (2022)
- [Exact Detection Thresholds and Minimax Optimality of Chatterjee's Correlation Coefficient](https://mlsanity.com/paper/2104.15140/) (2021)
- [On the asymptotic distribution of the symmetrized Chatterjee's correlation coefficient](https://mlsanity.com/paper/2205.01769/) (2022)
- [Measures of independence and functional dependence](https://mlsanity.com/paper/2206.13663/) (2022)
- [A direct extension of Azadkia & Chatterjee's rank correlation to multi-response vectors](https://mlsanity.com/paper/2212.01621/) (2022)
- [Association and Independence Test for Random Objects](https://mlsanity.com/paper/2505.01983/) (2025)
- [A new measure of dependence: Integrated R2](https://mlsanity.com/paper/2505.18146/) (2025)
- [Quantifying directed dependence via dimension reduction](https://mlsanity.com/paper/2112.10147/) (2021)
- [On relationships between Chatterjee's and Spearman's correlation coefficients](https://mlsanity.com/paper/2302.10131/) (2023)
- [Coverage correlation: detecting singular dependencies between random variables](https://mlsanity.com/paper/2508.06402/) (2025)
- [A multivariate extension of Azadkia-Chatterjee's rank coefficient](https://mlsanity.com/paper/2512.07443/) (2025)

## Abstract (arXiv)

We introduce kernel integrated R2, a new measure of statistical dependence that combines the local normalization principle of the recently introduced integrated R2 with the flexibility of reproducing kernel Hilbert spaces (RKHSs). The proposed measure extends integrated R2 from scalar responses to responses taking values on general spaces equipped with a characteristic kernel, allowing to measure dependence of multivariate, functional, and structured data, while remaining sensitive to tail behaviour and oscillatory dependence structures. We establish that (i) this new measure takes values in [0,1], (ii) equals zero if and only if independence holds, and (iii) equals one if and only if the response is almost surely a measurable function of the covariates. Two estimators are proposed: a graph-based method using K-nearest neighbours and an RKHS-based method built on conditional mean embeddings. We prove consistency and derive convergence rates for the graph-based estimator, showing its adaptation to intrinsic dimensionality. Numerical experiments on simulated data and a real data experiment in the context of dependency testing for media annotations demonstrate competitive power against state-of-the-art dependence measures, particularly in settings involving non-linear and structured relationships.