Relating Leverage Scores and Density using Regularized Christoffel Functions
This provides theoretical insight into a fundamental tool in machine learning, though it appears incremental as it builds on existing orthogonal polynomial literature.
The paper tackled the problem of understanding the relationship between statistical leverage scores and population density by introducing regularized Christoffel functions, revealing a quantitative decreasing relation for a broad class of kernels.
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.