标题： 无约束正交矩阵表示及其在共同主成分中的应用 显示英文标题

标题： Unconstrained representation of orthogonal matrices with application to common principle components

摘要： Many statistical problems involve the estimation of a $\left(d\times d\right)$ orthogonal matrix $\textbf{Q}$. Such an estimation is often challenging due to the orthonormality constraints on $\textbf{Q}$. To cope with this problem, we propose a very simple decomposition for orthogonal matrices which we abbreviate as PLR decomposition. It produces a one-to-one correspondence between $\textbf{Q}$ and a $\left(d\times d\right)$ unit lower triangular matrix $\textbf{L}$ whose $d\left(d-1\right)/2$ entries below the diagonal are unconstrained real values. 一旦应用了分解，无论考虑的是哪个目标函数，我们都可以使用任何经典的无约束优化方法来找到目标函数相对于$\textbf{L}$的最小值（或最大值）。 为了说明目的，我们在共同主成分分析（CPCA）中应用PLR分解，当每个组假设为多变量尖峰正态分布时，用于共同正交矩阵的最大似然估计。 与常用的正态分布相比，尖峰正态分布有一个额外的参数控制超额峰度；这使得在CPCA中估计$\textbf{Q}$更加稳健，能够抵御轻微异常值。 PLR分解在尖峰正态CPCA中的有效性通过两个生物特征数据分析得到了说明。 显示英文摘要

摘要： Many statistical problems involve the estimation of a $\left(d\times d\right)$ orthogonal matrix $\textbf{Q}$. Such an estimation is often challenging due to the orthonormality constraints on $\textbf{Q}$. To cope with this problem, we propose a very simple decomposition for orthogonal matrices which we abbreviate as PLR decomposition. It produces a one-to-one correspondence between $\textbf{Q}$ and a $\left(d\times d\right)$ unit lower triangular matrix $\textbf{L}$ whose $d\left(d-1\right)/2$ entries below the diagonal are unconstrained real values. Once the decomposition is applied, regardless of the objective function under consideration, we can use any classical unconstrained optimization method to find the minimum (or maximum) of the objective function with respect to $\textbf{L}$. For illustrative purposes, we apply the PLR decomposition in common principle components analysis (CPCA) for the maximum likelihood estimation of the common orthogonal matrix when a multivariate leptokurtic-normal distribution is assumed in each group. Compared to the commonly used normal distribution, the leptokurtic-normal has an additional parameter governing the excess kurtosis; this makes the estimation of $\textbf{Q}$ in CPCA more robust against mild outliers. The usefulness of the PLR decomposition in leptokurtic-normal CPCA is illustrated by two biometric data analyses.