The Annals of Statistics

Approximate optimal designs for multivariate polynomial regression

Yohann De Castro, Fabrice Gamboa, Didier Henrion, Roxana Hess, and Jean-Bernard Lasserre

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We introduce a new approach aiming at computing approximate optimal designs for multivariate polynomial regressions on compact (semialgebraic) 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.

Article information

Ann. Statist., Volume 47, Number 1 (2019), 127-155.

Received: June 2017
Revised: January 2018
First available in Project Euclid: 30 November 2018

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Mathematical Reviews number (MathSciNet)

Primary: 62K05: Optimal designs 90C25: Convex programming
Secondary: 41A10: Approximation by polynomials {For approximation by trigonometric polynomials, see 42A10} 49M29: Methods involving duality 90C90: Applications of mathematical programming 15A15: Determinants, permanents, other special matrix functions [See also 19B10, 19B14]

Experimental design semidefinite programming Christoffel polynomial linear model equivalence theorem


De Castro, Yohann; Gamboa, Fabrice; Henrion, Didier; Hess, Roxana; Lasserre, Jean-Bernard. Approximate optimal designs for multivariate polynomial regression. Ann. Statist. 47 (2019), no. 1, 127--155. doi:10.1214/18-AOS1683.

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Supplemental materials

  • Supplementary material of approximate optimal designs for multivariate polynomial regression. We provide the proof of Theorem 1 and the detailed numerical results of the numerical examples in supplement article [2].