Estimating the Spectral Moments of the Kernel Integral Operator from Finite Sample Matrices

TL;DR

This paper introduces an unbiased, efficient algorithm to estimate the spectral moments of the kernel integral operator from finite sample matrices, improving analysis of data structure and neural network representations.

Contribution

The paper presents a novel dynamic programming-based method for unbiased spectral moment estimation, addressing biases in traditional eigenvalue spectrum analysis from finite samples.

Findings

Accurate estimation of spectral moments for RBF kernels.

Method demonstrates consistency with theoretical spectra.

Robustness in analyzing neural network representations.

Abstract

Analyzing the structure of sampled features from an input data distribution is challenging when constrained by limited measurements in both the number of inputs and features. Traditional approaches often rely on the eigenvalue spectrum of the sample covariance matrix derived from finite measurement matrices; however, these spectra are sensitive to the size of the measurement matrix, leading to biased insights. In this paper, we introduce a novel algorithm that provides unbiased estimates of the spectral moments of the kernel integral operator in the limit of infinite inputs and features from finitely sampled measurement matrices. Our method, based on dynamic programming, is efficient and capable of estimating the moments of the operator spectrum. We demonstrate the accuracy of our estimator on radial basis function (RBF) kernels, highlighting its consistency with the theoretical spectra. Furthermore, we showcase the practical utility and robustness of our method in understanding the geometry of learned representations in neural networks.