Relating Leverage Scores and Density using Regularized Christoffel Functions

Edouard Pauwels (1) , Francis Bach (2) , Jean-Philippe Vert (3, 4, 5)

1 IRIT - Institut de recherche en informatique de Toulouse
2 SIERRA - Statistical Machine Learning and Parsimony
3 CBIO - Centre de Bioinformatique
4 Cancer et génome: Bioinformatique, biostatistiques et épidémiologie d'un système complexe
5 DMA - Département de Mathématiques et Applications - ENS Paris

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.

Domaines: Machine Learning [stat.ML]; Apprentissage [cs.LG]

Edouard Pauwels, Francis Bach, Jean-Philippe Vert. Relating Leverage Scores and Density using Regularized Christoffel Functions. 2018. ⟨hal-01796591v1⟩