Abstract: In this paper, we summarize the past, present of principle component analysis for tensors in the context of information compression, and show some untouched research fields. Firstly, we review the conception of tensors and tensor decomposition which can be expressed by an unified statistical model. Secondly, by the order of the classical principal component analysis, robust principal component analysis and sparse principal component analysis, we  summarize the development of relative statistical theories and algorithms where each one can be further divided into vector data, matrix data and tensor data from the simple to the complex.

Key words: principle component analysis for tensors weighted-CUSUM, compression theory for high-dimensional information, Tucker decomposition, robust PCA, sparse PCA