Nearest-Neighbor Radii under Dependent Sampling

TL;DR

This paper investigates how dependence in data affects the geometric properties of nearest-neighbor methods, providing theoretical guarantees and empirical evidence that their effectiveness persists under dependent sampling.

Contribution

It extends the analysis of nearest-neighbor radii to dependent data, establishing convergence and moment bounds that depend on local intrinsic dimension.

Findings

Nearest-neighbor geometry remains informative under dependent sampling.

Distribution-free convergence results hold for polynomial mixing.

Moment bounds depend on local intrinsic dimension, not ambient dimension.

Abstract

Nearest-neighbor methods are fundamental to classical and modern machine learning, yet their geometric properties are typically analyzed under independent sampling. In this paper, we study the nearest-neighbor radii under dependent sampling. We consider strong mixing dependent observations and ask whether dependence changes the scale of nearest-neighbor neighborhoods. We establish distribution-free almost sure convergence under polynomial mixing and sharp non-asymptotic moment bounds under geometric mixing. The moment bounds depend on the local intrinsic dimension rather than the ambient dimension, making the results applicable to high-dimensional data concentrated near lower-dimensional manifolds. Synthetic experiments and real-world time-series benchmarks support the theory, showing that nearest-neighbor geometry remains informative under dependence sampling.