Relating Leverage Scores and Density using Regularized Christoffel Functions

Abstract

Statistical leverage scores emerged as a fundamental tool for matrix sketching and column sampling with applications to low rank approximation, regression, random feature learning and quadrature. Yet, the very nature of this quantity is barely understood. Borrowing ideas from the orthogonal polynomial literature, we introduce the regularized Christoffel function associated to a positive definite kernel. This uncovers a variational formulation for leverage scores for kernel methods and allows to elucidate their relationships with the chosen kernel as well as population density. Our main result quantitatively describes a decreasing relation between leverage score and population density for a broad class of kernels on Euclidean spaces. Numerical simulations support our findings.1 Kernelized Christoffel functions were first proposed by Laurent El Ghaoui and independently studied in [4].32nd Conference on Neural Information Processing Systems (NIPS 2018),

Perturbations of Christoffel–Darboux Kernels: Detection of Outliers

“…and a combination with (26) for w = z gives (32). Moreover, it follows from (35) that M n (µ, ν) −1 − I ≤ , and thus, again by (26), (33).…”

“…The aim of this section is to recall some links between the Christoffel-Darboux kernel and several notions from statistics. For a recent related work see [32]. In this subsection we will denote elements of C d as row vectors.…”