Block Principal Component Analysis (Block PCA) of a data matrix A, where loadings Z are determined by maximization of AZ 2 over unit norm orthogonal loadings, is difficult to use for the design of sparse PCA by 1 regularization, due to the difficulty of taking care of both the orthogonality constraint on loadings and the non differentiable 1 penalty. Our objective in this paper is to relax the orthogonality constraint on loadings by introducing new objective functions expvar(Y) which measure the part of the variance of the data matrix A explained by correlated components Y = AZ. So we propose first a comprehensive study of mathematical and numerical properties of expvar(Y) for two existing definitions Zou et al. [2006], Shen and Huang [2008] and four new definitions. Then we show that only two of these explained variance are fit to use as objective function in block PCA formulations for A rid of orthogonality constraints.

翻译：数据矩阵A的块主成分分析（Block PCA）中，载荷Z通过最大化单位范数正交载荷下的∥AZ∥²来确定。由于需要兼顾载荷的正交约束和不可微的ℓ1惩罚项，该方法难以用于通过ℓ1正则化设计稀疏PCA。本文旨在通过引入新的目标函数expvar(Y)来放松载荷的正交约束，该函数衡量由相关成分Y=AZ解释的数据矩阵A的方差部分。为此，我们首先对两种现有定义（Zou等人[2006]，Shen与Huang[2008]）及四种新定义的expvar(Y)的数学与数值性质进行全面研究。随后证明，在无正交约束的A的块PCA框架中，仅有两种解释方差适合作为目标函数使用。