# Nearest neighbor empirical processes

arXiv: https://arxiv.org/abs/2110.15083 · HTML: https://mlsanity.com/paper/2110.15083/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- Builds on 5 of its 64 references that are in the map.
- First posted on arXiv 27 Oct 2021.

## Builds on (most cited first, 5 of 5)

- [Hierarchical Nearest-Neighbor Gaussian Process Models for Large Geostatistical Datasets](https://mlsanity.com/paper/1406.7343/) (2014)
- [Local nearest neighbour classification with applications to semi-supervised learning](https://mlsanity.com/paper/1704.00642/) (2017)
- [Nearest Neighbour Based Estimates of Gradients: Sharp Nonasymptotic Bounds and Applications](https://mlsanity.com/paper/2006.15043/) (2020)
- [Rates of Convergence for Large-scale Nearest Neighbor Classification](https://mlsanity.com/paper/1909.01464/) (2019)
- [Risk bounds when learning infinitely many response functions by ordinary linear regression](https://mlsanity.com/paper/2006.09223/) (2020)

## Built on by (most cited first, 2 of 2)

- [Concentration and excess risk bounds for imbalanced classification with synthetic oversampling](https://mlsanity.com/paper/2510.20472/) (2025)
- [Conditional Mean and Variance Estimation via textitk-NN Algorithm with Automated Variance Selection](https://mlsanity.com/paper/2402.01635/) (2024)

## Abstract (arXiv)

In the regression framework, the empirical measure based on the responses resulting from the nearest neighbors, among the covariates, to a given point x is introduced and studied as a central statistical quantity. First, the associated empirical process is shown to satisfy a uniform central limit theorem under a local bracketing entropy condition on the underlying class of functions reflecting the localizing nature of the nearest neighbor algorithm. Second a uniform non-asymptotic bound is established under a well-known condition, often referred to as Vapnik-Chervonenkis, on the uniform entropy numbers. The covariance of the Gaussian limit obtained in the uniform central limit theorem is simply equal to the conditional covariance operator given the covariate value. This suggests the possibility of using standard formulas to estimate the variance by using only the nearest neighbors instead of the full data. This is illustrated on two problems: the estimation of the conditional cumulative distribution function and local linear regression.