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

Ching-Shui Cheng

doi:10.1214/aos/1176350612

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

Ching-Shui Cheng刊期 1987-12-01 · 卷 15（4）The Annals of Statistics已编入正刊

Kiefer–Wolfowitz 等价定理在哈达玛变换光学问题中的应用AI 解读 ↓

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被引 32 次 ↗OpenAlex · 核验 2026-10-04

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DOI · 10.1214/aos/1176350612

The Annals of Statistics · 卷 15，期 4 · 页码 / 文章号 1593-1603

正式刊期日期：1987-12-01

DOI 首次登记：2007-12-17

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Ching-Shui Cheng

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Kiefer–Wolfowitz 等价定理在哈达玛变换光学问题中的应用

研究问题: 在哈达玛变换光谱学的矩阵设计问题中，如何证明相关最优性猜想，并求解更一般的信息矩阵最优设计问题？
方法与思路: 作者将矩阵优化推广为单位立方体上概率测度的设计优化，使用最优设计等价定理，并借助平衡不完全区组设计缩减支撑。
材料陈述的贡献: 摘要称，广义设计问题得到完整解，最优测度可由特定顶点上的均匀分布或相邻层混合构成，并讨论奇偶维数情形。
阅读关注点: 应核对设计准则、参数取值范围、矩阵可逆性与区组设计存在条件，以及连续测度最优性如何转化为有限矩阵结论。
阅读边界: OpenAlex 摘要版本未注明，部分指数、括号和参数上界表达疑有转录问题，须查原 PDF；不能据此重构公式或推导精确范围。

GPT-6.1 Sol · 高 · 2026-10-07

原文摘要

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.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.

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