Optimal Designs for Regression on Lie Groups

We consider a linear regression model with complex-valued response and predictors from a compact and connected Lie group. The regression model is formulated in terms of eigenfunctions of the Laplace-Beltrami operator on the Lie group. We show that the normalized Haar measure is an approximate optimal design with respect to all Kiefer's $\Phi_p$-criteria. Inspired by the concept of $t$-designs in the field of algebraic combinatorics, we then consider so-called $\lambda$-designs in order to construct exact $\Phi_p$-optimal designs for fixed sample sizes in the considered regression problem. In particular, we explicitly construct $\Phi_p$-optimal designs for regression models with predictors in the Lie groups $\mathrm{SU}(2)$ and $\mathrm{SO}(3)$, the groups of $2\times 2$ unitary matrices and $3\times 3$ orthogonal matrices with determinant equal to $1$, respectively. We also discuss the advantages of the derived theoretical results in a concrete biological application.