Approximate D-optimal design and equilibrium measure

TL;DR

This paper introduces a variant of D-optimal design that leverages equilibrium measures for specific geometric sets, providing closed-form solutions and convergence results, with applications to cubature and semi-algebraic sets.

Contribution

It establishes a connection between approximate D-optimal design and equilibrium measures, enabling closed-form solutions for certain sets and extending to semi-algebraic sets.

Findings

Closed-form solutions for D-optimal design on specific sets

Convergence of atomic D-optimal measures to equilibrium measures

Extension of design algorithms to semi-algebraic sets

Abstract

We introduce a minor variant of the approximate D-optimal design of experiments with a more general information matrix that takes into account the representation of the design space S. The main motivation (and result) is that if S in R^d is the unit ball, the unit box or the canonical simplex, then remarkably, for every dimension d and every degree n, one obtains an optimal solution in closed form, namely the equilibrium measure of S (in pluripotential theory). Equivalently, for each degree n, the unique optimal solution is the vector of moments (up to degree 2n) of the equilibrium measure of S. Hence finding an optimal design reduces to finding a cubature for the equilibrium measure, with atoms in S, positive weights, and exact up to degree 2n. In addition, any resulting sequence of atomic D-optimal measures converges to the equilibrium measure of S for the weak-star topology, as n increases. Links with Fekete sets of points are also discussed. More general compact basic semi-algebraic sets are also considered, and a previously developed two-step design algorithm is easily adapted to this new variant of D-optimal design problem.