# Nystr\"om landmark sampling and regularized Christoffel functions

## TL;DR

This paper introduces new deterministic and randomized algorithms for selecting diverse landmark points in large datasets, leveraging kernelized Christoffel functions and connections to DPPs to improve kernel methods like Ridge Regression.

## Contribution

It proposes novel adaptive algorithms for landmark selection based on Christoffel functions, linking them to DPPs and enhancing kernel approximation and regression accuracy.

## Key findings

- Algorithms promote diversity among landmarks.
- Improved kernel approximation via Christoffel functions.
- Enhanced accuracy in Kernel Ridge Regression.

## Abstract

Selecting diverse and important items, called landmarks, from a large set is a problem of interest in machine learning. As a specific example, in order to deal with large training sets, kernel methods often rely on low rank matrix Nystr\"om approximations based on the selection or sampling of landmarks. In this context, we propose a deterministic and a randomized adaptive algorithm for selecting landmark points within a training data set. These landmarks are related to the minima of a sequence of kernelized Christoffel functions. Beyond the known connection between Christoffel functions and leverage scores, a connection of our method with finite determinantal point processes (DPPs) is also explained. Namely, our construction promotes diversity among important landmark points in a way similar to DPPs. Also, we explain how our randomized adaptive algorithm can influence the accuracy of Kernel Ridge Regression.