Estimating the Spectral Moments of the Kernel Integral Operator from Finite Sample Matrices
This addresses the challenge of analyzing feature structures with limited data for researchers in kernel methods and representation learning, though it is incremental as it builds on existing spectral estimation techniques.
The paper tackles the problem of biased eigenvalue spectra from finite sample covariance matrices by introducing a novel algorithm that provides unbiased estimates of the spectral moments of the kernel integral operator from finite measurements, demonstrating accuracy on RBF kernels and utility in analyzing neural network representations.
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.