研究依赖采样下近邻半径的几何性质，发现其仍具信息量。
Nearest-Neighbor Radii under Dependent Sampling
- 在强混合依赖数据下分析近邻半径的收敛性
- 给出多项式混合下的几乎必然收敛与几何混合下的精确矩界
- 适用于高维流形数据，对时间序列等场景有指导意义
近邻方法是经典与现代机器学习的基础，但其几何性质通常在独立采样假设下分析。本文研究依赖采样下的近邻半径。考虑强混合依赖观测，探究依赖是否改变近邻邻域尺度。在多项式混合下建立分布无关的几乎必然收敛，在几何混合下获得紧致的非渐近矩界。矩界依赖于局部内在维度而非环境维度，使结果适用于集中在低维流形上的高维数据。合成实验与真实时间序列基准验证理论，表明依赖采样下近邻几何依然具有信息性。
原文摘要 · Abstract (English)
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.
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