Dual optimal design and the Christoffel-Darboux polynomial

TL;DR

This paper reveals that the Christoffel-Darboux polynomial naturally emerges as the dual solution in semi-algebraic D-optimal experimental design, connecting approximation theory and data science with elementary convex analysis.

Contribution

It demonstrates the geometric and algorithmic significance of the Christoffel-Darboux polynomial in optimal experimental design, providing new insights into its foundational role.

Findings

Christoffel-Darboux polynomial is the dual of semi-algebraic D-optimal design

Elementary convex analysis explains the polynomial's emergence

Implications for geometric interpretation and algorithms in design

Abstract

The purpose of this short note is to show that the Christoffel-Darboux polynomial, useful in approximation theory and data science, arises naturally when deriving the dual to the problem of semi-algebraic D-optimal experimental design in statistics. It uses only elementary notions of convex analysis. Geometric interpretations and algorithmic consequences are mentioned.