半正规化Hadamard矩阵的概率构造与分析
Probabilistic Construction and Analysis of Seminormalized Hadamard Matrices
摘要：
设o是一个长度为4k的列向量，其所有条目都是1，k是正整数。设V={v_i}是一组半规范化Hadamard（SH）向量，它们是4k长度的向量，其2k个条目是-1 s，其余2k个是1 s。我们定义一个4k阶QSH（Quasi SH）-矩阵Q为4kx 4k矩阵，其中第一列为o，其余列在V中不同v_i。当Q正交时，它成为SH-矩阵H。因此，可以通过从v_i的每一个组合中枚举所有可能的Q，然后评估它们中的每一个的正交性来构造4k阶SH-矩阵。由于这种穷举方法需要大量的计算资源，我们可以采用概率算法来构造H，例如，通过随机向量选择（RVS）或模拟退火（OSA）算法的归一化。利用这些方法，我们证明了低阶SH-矩阵的构造。我们还分析了该构造的一些概率方面，包括随机选取的SH-向量对之间的正交概率p，存在概率p_H| Q，随机生成的Q实际上是SH-矩阵H，并解决了SH-矩阵的已知数量与从概率分析导出的期望数量之间的分布差异。
英文摘要：
Let o be a 4k-length column vector whose all entries are 1s, with k a positive integer. Let V={v_i} be a set of semi-normalized Hadamard (SH)-vectors, which are 4k-length vectors whose 2k entries are -1s and the remaining 2k are 1s. We define a 4k-order QSH (Quasi SH)-matrix, Q, as a 4kx4k matrix where the first column is o and the remaining ones are distinct v_i in V. When Q is orthogonal, it becomes an SH-matrix H. Therefore, 4k-order SH-matrices can be built by enumerate all possible Q from every combination of v_i, then evaluate the orthogonality of each one of them. Since such exhaustive method requires a large amount of computing resource, we can employ probabilistic algorithms to construct H, such as, by Random Vector Selection (RVS) or the Orthogonalization by Simulated Annealing (OSA) algorithms. We demonstrate the constructions of low-order SH-matrices by using these methods. We also analyze some probabilistic aspects of the constructions, including orthogonal probability p between a pair of randomly selected SH-vectors, the existence probability p_H|Q that a randomly generated Q is in fact an SH-matrix H, and address the discrepancy of the distribution between the known number of SH-matrix with expected number derived from the probabilistic analysis.
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