An Application of the Kiefer-Wolfowitz Equivalence Theorem to a Problem in Hadamard Transform Optics

Abstract

Let $\Omega = \{\mathbf{x} = (x_1, \cdots, x_n)^T: 0 \leq x_i \leq 1\}$ be the unit cube in $R^n$. For any probability measure $\xi$ on $\Omega$, let $\mathbf{M}(\xi) = \int_\Omega\mathbf{xx}^T \xi(d\mathbf{x})$. Harwit and Sloane (1976) conjectured that if $\mathbf{X}^\ast$ is the incidence matrix of a balanced incomplete block design (BIBD) with $n$ treatments and $n$ blocks of size $(n + 1)/2$, then $\mathbf{X}^\ast$ minimizes $\operatorname{tr}(\mathbf{X}^T\mathbf{X})^{-1}$ over the $n \times n$ matrices with entries $0 \leq x_{ij} \leq 1$. This arises from a problem in spectroscopy. In order to solve the conjecture, we consider the more general problem of maximizing $j_a(\mathbf{M}(\xi))$ over the probability measures on $\Omega$ for $-\infty \leq a \leq 1$, where $j_0 (\mathbf{M}(\xi)) = \{\det \mathbf{M}(\xi))\}^{1/n}, j_{-\infty} (\mathbf{M}(\xi)) =$ the minimum eigenvalue of $\mathbf{M}(\xi)$ and $j_a(\mathbf{M}(\xi)) = \{n^{-1} \operatorname{tr}\lbrack \mathbf{M}(\xi) \rbrack^a\}^{1/a}$ for other $a$'s. A complete solution is obtained by using the equivalence theorem in optimal design theory. Let $\xi_k$ be the uniform measure on the vertices of $\Omega$ with $k$ coordinates equal to 1. Then depending on the value of $a$, optimality is attained by $\xi_k$ or a mixture of $\xi_k$ and $\xi_{k+1}$ with $k \geq \lbrack (n + 1)/2\rbrack$. Optimal $\xi$'s with a smaller support can be found by using BIBDs. It follows that if $n$ is odd and $\mathbf{X}^\ast$ is the block-treatment incidence matrix of a BIBD with $n$ treatments and $N$ blocks of size $(n + 1)/2$, then $\mathbf{X}^\ast$ minimizes $\operatorname{tr}(\mathbf{X}^T\mathbf{X})^a$ for all $a < 0$ and maximizes $\det(\mathbf{X}^T\mathbf{X})^a$ and $\operatorname{tr}(\mathbf{X}^T\mathbf{X})^a$ for all $0 < a \leq 1 - \ln(n/2 + 1)/ \ln(n/2 + 1)/ \ln( n + 1)$ over the $N \times n$ matrices with entries $0 \leq x_{ij} \leq 1$. Similar results are derived for the even case and the incidence matrices of BIBDs of larger block sizes.