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.

通过引入新的目标函数，衡量数据矩阵 A 相关分量 Y=AZ 所解释的方差部分，本文在对已有定义的解释方差的数学和数值特性进行全面研究的基础上，提出了两个最适合用作消除正交性约束的 block PCA 公式的目标函数。