Nearest-Neighbor Radii under Dependent Sampling

2026-05-14

AI Summary

This study investigates how dependence in sampling—such as that arising in time series—affects the geometric properties of nearest neighbor methods, with a focus on the scaling behavior of the nearest neighbor radius. For strongly mixing dependent observations, the work establishes, for the first time, distribution-free almost sure convergence and non-asymptotic moment bounds. The theoretical analysis integrates strong mixing process modeling, probabilistic inequalities, and geometric tools, revealing that these moment bounds are governed by the local intrinsic dimensionality rather than the ambient dimension, making the results particularly relevant for high-dimensional manifold-structured data. Empirical validation on both synthetic datasets and real-world time series benchmarks confirms the theoretical findings, demonstrating that nearest neighbor geometry retains meaningful information even under dependent sampling.

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.

Problem

• nearest-neighbor radii
• dependent sampling
• strong mixing
• geometric properties
• nearest-neighbor methods

Innovation

• nearest-neighbor radii
• dependent sampling
• strong mixing
• intrinsic dimension
• non-asymptotic bounds