Approximate Optimal Designs for Multivariate Polynomial Regression

Yohann de Castro 1, 2 Fabrice Gamboa 3 Didier Henrion 4, 5 Roxana Hess 4 Jean Bernard Lasserre 4
1 LMO - Laboratoire de Mathématiques d'Orsay
2 MOKAPLAN - Méthodes numériques pour le problème de Monge-Kantorovich et Applications en sciences sociales
Inria de Paris, CEREMADE - CEntre de REcherches en MAthématiques de la DEcision
3 IMT - Institut de Mathématiques de Toulouse UMR5219
4 LAAS-MAC - Équipe Méthodes et Algorithmes en Commande
LAAS - Laboratoire d'analyse et d'architecture des systèmes
5 CTU - Czech Technical University in Prague
Abstract : 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.
Keywords : Semidefinite programming Christoffel Polynomial Experimental design Linear Model Equivalence Theorem
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https://hal.laas.fr/hal-01483490
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Submitted on : Tuesday, October 17, 2017 - 10:41:26 AM
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CNRS | INSMI | TDS-MACS | IMT | UNIV-TLSE2 | INSA-TOULOUSE | LAAS | UNIV-PARIS-SACLAY | INRIA | LAAS-DECISION-ET-OPTIMISATION | LAAS-MAC | INSA-GROUPE | CEREMADE | PSL | UNIV-TLSE3 | UT1-CAPITOLE | UNIV-DAUPHINE

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Yohann de Castro, Fabrice Gamboa, Didier Henrion, Roxana Hess, Jean Bernard Lasserre. Approximate Optimal Designs for Multivariate Polynomial Regression. Annals of Statistics, Institute of Mathematical Statistics, 2019, 47 (1), pp.127-155. ⟨10.1214/18-AOS1683⟩. ⟨hal-01483490v2⟩

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