标题： 基于Procrustes重定标的正交稀疏PCA和协方差估计 显示英文标题

标题： Orthogonal Sparse PCA and Covariance Estimation via Procrustes Reformulation

摘要： 估计对称矩阵稀疏特征向量的问题在许多应用中引起了广泛关注，特别是在高维数据集的情况下。 虽然经典特征向量可以通过一个最大化问题的解来获得，但现有的方法通过在目标函数中添加一个鼓励稀疏解的惩罚项来构建该问题。 然而，这些方法以牺牲正交性为代价来实现稀疏性。 本文提出了一种新的方法，用于估计主导稀疏特征向量而不牺牲它们的正交性。 该问题高度非凸且难以处理。我们采用MM框架，在Stiefel流形上迭代最大化目标函数的一个紧密下界（替代函数）。 内层的最大化问题最终归结为一个矩形Procrustes问题，该问题具有闭式解。 此外，我们还提出了一种方法来改进协方差估计问题，当其底层特征向量已知为稀疏时。 我们利用协方差矩阵的特征值分解来构建一个优化问题，在对应的特征向量上施加稀疏性约束。 数值实验表明，所提出的特征向量提取算法在支持恢复和解释方差方面与现有算法相当或优于现有算法，而协方差估计算法显著改进了样本协方差估计器。 显示英文摘要

摘要： The problem of estimating sparse eigenvectors of a symmetric matrix attracts a lot of attention in many applications, especially those with high dimensional data set. While classical eigenvectors can be obtained as the solution of a maximization problem, existing approaches formulate this problem by adding a penalty term into the objective function that encourages a sparse solution. However, the resulting methods achieve sparsity at the expense of sacrificing the orthogonality property. In this paper, we develop a new method to estimate dominant sparse eigenvectors without trading off their orthogonality. The problem is highly non-convex and hard to handle. We apply the MM framework where we iteratively maximize a tight lower bound (surrogate function) of the objective function over the Stiefel manifold. The inner maximization problem turns out to be a rectangular Procrustes problem, which has a closed form solution. In addition, we propose a method to improve the covariance estimation problem when its underlying eigenvectors are known to be sparse. We use the eigenvalue decomposition of the covariance matrix to formulate an optimization problem where we impose sparsity on the corresponding eigenvectors. Numerical experiments show that the proposed eigenvector extraction algorithm matches or outperforms existing algorithms in terms of support recovery and explained variance, while the covariance estimation algorithms improve significantly the sample covariance estimator.