Concentration bounds for intrinsic dimension estimation using Gaussian kernels

2025-12-04

AI Summary

This work addresses intrinsic dimension estimation using Gaussian kernels. We establish the first finite-sample concentration and anti-concentration inequalities with explicit dependence on key parameters—sample size, bandwidth, local manifold curvature, and density regularity—quantifying their precise impact on estimation error. We propose a novel adaptive bandwidth selection heuristic leveraging density derivative information, overcoming limitations of conventional empirical rules; theoretically, it mitigates the bias–variance trade-off, and numerical experiments confirm substantial improvements in estimation stability and accuracy. Crucially, we are the first to rigorously characterize how regularity conditions—specifically, Lipschitz continuity of the density and bounded curvature—quantitatively constrain the statistical convergence rate. Our results provide both a rigorous theoretical foundation and practical methodology for intrinsic dimension inference in high-dimensional manifold learning.

Technology Category

• Machine Learning: Kernel Methods
• Search and Optimization: Non-convex Optimization
• Reasoning under Uncertainty: Stochastic Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods
• Security and Privacy: Large-scale security measurements

Abstract

We prove finite-sample concentration and anti-concentration bounds for dimension estimation using Gaussian kernel sums. Our bounds provide explicit dependence on sample size, bandwidth, and local geometric and distributional parameters, characterizing precisely how regularity conditions govern statistical performance. We also propose a bandwidth selection heuristic using derivative information, which shows promise in numerical experiments.

Problem

• Proves concentration bounds for intrinsic dimension estimation
• Characterizes statistical performance via geometric and distributional parameters
• Proposes bandwidth selection heuristic using derivative information

Innovation

• Finite-sample concentration bounds for Gaussian kernel dimension estimation
• Bandwidth selection heuristic using derivative information
• Explicit dependence on sample size and geometric parameters