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 $S\subset \mathbb{R}^{d}$ is the unit ball, the unit box or the canon...
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| Main Authors: | , |
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| Format: | Article |
| Language: | English |
| Published: |
Académie des sciences
2025-06-01
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| Series: | Comptes Rendus. Mathématique |
| Online Access: | https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.766/ |
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| Summary: | 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 $S\subset \mathbb{R}^{d}$ 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. |
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| ISSN: | 1778-3569 |