A general technique for approximating high-dimensional empirical kernel matrices

2025-11-05

AI Summary

This work addresses the problem of deriving tight approximations to the operator norm of empirical kernel matrices in high dimensions, with a focus on inner-product kernels and anisotropic Gaussian data, and establishes lower bounds on the bias of kernel regression. We propose a unified analytical framework that integrates decoupling of U-statistics, the noncommutative Khintchine inequality, and the moment method—yielding concise upper and lower bounds on the expected operator norm solely in terms of low-order scalar statistics of the kernel function and its associated kernel matrix. Compared to prior approaches, our method significantly simplifies proofs and, for the first time, extends tight norm bounds to the anisotropic Gaussian setting. Under a high-dimensional regime where sample size and dimension scale polynomially, our bounds are strictly tighter, leading to improved lower bounds on kernel regression bias. This provides both a novel theoretical tool and a new benchmark for analyzing high-dimensional kernel methods.

Technology Category

• Machine Learning: Kernel Methods
• Reasoning under Uncertainty: Stochastic Optimization
• Search and Optimization: Non-convex 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
• Economics, Online Markets and Human Computation: Data quality aspects of human-annotated datasets

Abstract

We present simple, user-friendly bounds for the expected operator norm of a random kernel matrix under general conditions on the kernel function $k(cdot,cdot)$. Our approach uses decoupling results for U-statistics and the non-commutative Khintchine inequality to obtain upper and lower bounds depending only on scalar statistics of the kernel function and a ``correlation kernel''matrix corresponding to $k(cdot,cdot)$. We then apply our method to provide new, tighter approximations for inner-product kernel matrices on general high-dimensional data, where the sample size and data dimension are polynomially related. Our method obtains simplified proofs of existing results that rely on the moment method and combinatorial arguments while also providing novel approximation results for the case of anisotropic Gaussian data. Finally, using similar techniques to our approximation result, we show a tighter lower bound on the bias of kernel regression with anisotropic Gaussian data.

Problem

• Approximating high-dimensional empirical kernel matrices
• Providing bounds for random kernel matrix operator norms
• Analyzing kernel regression bias with anisotropic Gaussian data

Innovation

• U-statistics decoupling for kernel matrix bounds
• Non-commutative Khintchine inequality application
• Correlation kernel matrix scalar statistics analysis