arXiv subjects

Mona Azadkia

Publications and source records attributed to Mona Azadkia.

10 recordsLinked to original sources

arXiv2026

Kernel Integrated $R^2$: A Measure of Dependence

We introduce kernel integrated $R^2$, a new measure of statistical dependence that combines the local normalization principle of the recently introduced integrated $R^2$ with the flexibility of reproducing kernel Hilbert spaces (RKHSs). The proposed measure extends integrated $R^2$ 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.

stat.ML↗

arXiv2026

Kernel Estimation Of Chatterjee's Dependence Coefficient

Dette, Siburg, and Stoimenov (2013) introduced a copula-based measure of dependence, which implies independence if it vanishes and is equal to 1 if one variable is a measurable function of the other. For continuous distributions, the dependence measure also appears as stochastic limit of Chatterjee's rank correlation (Chatterjee, 2021). They proved asymptotic normality of a corresponding kernel estimator with a parametric rate of convergence. In recent work Shi, Drton, and Han (2022) revealed empirically and theoretically that under independence the asymptotic variance degenerates. In this note, we derive the correct asymptotic distribution of the kernel estimator under the null hypothesis of independence. We show that after a suitable centering and rescaling at a rate larger than $\sqrt{n}$ (where $n$ is the sample size), the estimator is asymptotically normal. The analysis relies on a refined central limit theorem for double-indexed linear permutation statistics and accounts for boundary effects that are asymptotically non-negligible. As a consequence, we obtain a valid basis for independence testing without relying on permutations and argue that tests based on the kernel estimator detect local alternatives converging to the null at a faster rate than those detectable by Chatterjee's rank correlation.

math.ST↗

arXiv2025

Bias correction for Chatterjee's graph-based correlation coefficient

Azadkia and Chatterjee (2021) recently introduced a simple nearest neighbor (NN) graph-based correlation coefficient that consistently detects both independence and functional dependence. Specifically, it approximates a measure of dependence that equals 0 if and only if the variables are independent, and 1 if and only if they are functionally dependent. However, this NN estimator includes a bias term that may vanish at a rate slower than root-$n$, preventing root-$n$ consistency in general. In this article, we (i) analyze this bias term closely and show that it could become asymptotically negligible when the dimension is smaller than four; and (ii) propose a bias-correction procedure for more general settings. In both regimes, we obtain estimators (either the original or the bias-corrected version) that are root-$n$ consistent and asymptotically normal.

stat.ME↗

arXiv2025

Coverage correlation: detecting singular dependencies between random variables

We introduce the coverage correlation coefficient, a novel nonparametric measure of statistical association designed to quantify the extent to which two random variables have a joint distribution concentrated on a singular subset with respect to the product of the marginals. Our correlation statistic consistently estimates an $f$-divergence between the joint distribution and the product of the marginals, which is 0 if and only if the variables are independent and 1 if and only if the copula is singular. Using Monge--Kantorovich ranks, the coverage correlation naturally extends to measure association between random vectors. It is distribution-free, admits an analytically tractable asymptotic null distribution, and can be computed efficiently, making it well-suited for detecting complex, potentially nonlinear associations in large-scale pairwise testing.

stat.ME↗

arXiv2025

A new measure of dependence: Integrated $R^2$

We introduce a novel measure of dependence that captures the extent to which a random variable $Y$ is determined by a random vector $X$. The measure equals zero precisely when $Y$ and $X$ are independent, and it attains one exactly when $Y$ is almost surely a measurable function of $X$. We further extend this framework to define a measure of conditional dependence between $Y$ and $X$ given $Z$. We propose a simple and interpretable estimator with computational complexity comparable to classical correlation coefficients, including those of Pearson, Spearman, and Chatterjee. Leveraging this dependence measure, we develop a tuning-free, model-agnostic variable selection procedure and establish its consistency under appropriate sparsity conditions. Extensive experiments on synthetic and real datasets highlight the strong empirical performance of our methodology and demonstrate substantial gains over existing approaches.

math.ST↗

arXiv2023

Linear regression with unmatched data: a deconvolution perspective

Consider the regression problem where the response $Y\in\mathbb{R}$ and the covariate $X\in\mathbb{R}^d$ for $d\geq 1$ are \textit{unmatched}. Under this scenario, we do not have access to pairs of observations from the distribution of $(X, Y)$, but instead, we have separate datasets $\{Y_i\}_{i=1}^n$ and $\{X_j\}_{j=1}^m$, possibly collected from different sources. We study this problem assuming that the regression function is linear and the noise distribution is known or can be estimated. We introduce an estimator of the regression vector based on deconvolution and demonstrate its consistency and asymptotic normality under an identifiability assumption. In the general case, we show that our estimator (DLSE: Deconvolution Least Squared Estimator) is consistent in terms of an extended $\ell_2$ norm. Using this observation, we devise a method for semi-supervised learning, i.e., when we have access to a small sample of matched pairs $(X_k, Y_k)$. Several applications with synthetic and real datasets are considered to illustrate the theory.

math.ST↗

arXiv2022

A Fast Non-parametric Approach for Local Causal Structure Learning

We study the problem of causal structure learning with essentially no assumptions on the functional relationships and noise. We develop DAG-FOCI, a computationally fast algorithm for this setting that is based on the FOCI variable selection algorithm in~\cite{azadkia2021simple}. DAG-FOCI outputs the set of parents of a response variable of interest. We provide theoretical guarantees of our procedure when the underlying graph does not contain any (undirected) cycle containing the response variable of interest. Furthermore, in the absence of this assumption, we give a conservative guarantee against false positive causal claims when the set of parents is identifiable. We demonstrate the applicability of DAG-FOCI on simulated as well as a real dataset from computational biology~\cite{sachs2005causal}.

math.ST↗

arXiv2021

A simple measure of conditional dependence

We propose a coefficient of conditional dependence between two random variables $Y$ and $Z$ given a set of other variables $X_1,\ldots,X_p$, based on an i.i.d. sample. The coefficient has a long list of desirable properties, the most important of which is that under absolutely no distributional assumptions, it converges to a limit in $[0,1]$, where the limit is $0$ if and only if $Y$ and $Z$ are conditionally independent given $X_1,\ldots,X_p$, and is $1$ if and only if $Y$ is equal to a measurable function of $Z$ given $X_1,\ldots,X_p$. Moreover, it has a natural interpretation as a nonlinear generalization of the familiar partial $R^2$ statistic for measuring conditional dependence by regression. Using this statistic, we devise a new variable selection algorithm, called Feature Ordering by Conditional Independence (FOCI), which is model-free, has no tuning parameters, and is provably consistent under sparsity assumptions. A number of applications to synthetic and real datasets are worked out.

math.ST↗

arXiv2020

Optimal choice of $k$ for $k$-nearest neighbor regression

The $k$-nearest neighbor algorithm ($k$-NN) is a widely used non-parametric method for classification and regression. We study the mean squared error of the $k$-NN estimator when $k$ is chosen by leave-one-out cross-validation (LOOCV). Although it was known that this choice of $k$ is asymptotically consistent, it was not known previously that it is an optimal $k$. We show, with high probability, the mean squared error of this estimator is close to the minimum mean squared error using the $k$-NN estimate, where the minimum is over all choices of $k$.

math.ST↗

arXiv2019

Adaptive Estimation of Noise Variance and Matrix Estimation via USVT Algorithm

We propose a method for estimating the entries of a large noisy matrix when the variance of the noise, $σ^2$, is unknown without putting any assumption on the rank of the matrix. We consider the estimator for $σ$ introduced by Gavish and Donoho \cite{Gavish} and give an upper bound on its mean squared error. Then with the estimate of the variance, we use a modified version of the Universal Singular Value Thresholding (USVT) algorithm introduced by Chatterjee \cite{Chatterjee} to estimate the noisy matrix. Finally, we give an upper bound on the mean squared error of the estimated matrix.

math.ST↗