Approximate Optimal Designs for Multivariate Polynomial Regression
For statisticians and experimental designers, this provides a numerical method for optimal design in multivariate polynomial regression, though it is an incremental extension of existing semidefinite programming techniques.
The paper introduces a moment-sum-of-squares hierarchy to compute approximate optimal designs for multivariate polynomial regression, proving convergence and providing a dual certificate for finite convergence. The method links the equivalence theorem to Christoffel polynomials.
We introduce a new approach aiming at computing approximate optimal designs for multivariate polynomial regressions on compact (semi-algebraic) design spaces. We use the moment-sum-of-squares hierarchy of semidefinite programming problems to solve numerically the approximate optimal design problem. The geometry of the design is recovered via semidefinite programming duality theory. This article shows that the hierarchy converges to the approximate optimal design as the order of the hierarchy increases. Furthermore, we provide a dual certificate ensuring finite convergence of the hierarchy and showing that the approximate optimal design can be computed numerically with our method. As a byproduct, we revisit the equivalence theorem of the experimental design theory: it is linked to the Christoffel polynomial and it characterizes finite convergence of the moment-sum-of-square hierarchies.