SCIENTIA SINICA Informationis, Volume 48, Issue 12: 1622-1633(2018) https://doi.org/10.1360/N112017-00238

Information compression based on principal component analysis: from one-order to higher-order

• AcceptedJan 10, 2018
• PublishedSep 19, 2018
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Abstract

In this paper, a statistical technique of principal component analysis (PCA) from one-order to higher-order data under the background of information compression is summarized, its characteristics and limitations of each order PCA are revealed from three different perspectives, and some possible research directions are pointed out. Firstly, the technique and some existing developments are summarized by a similar structure, their intrinsic similar statistical significance is further analyzed, and the common structure—the Pythagorean theorem that shows two equivalent expressions of PCA—“maximizing variability" and “minimizing square loss" is shown. Secondly, this paper analyzes three important angles of PCA: the first one starts from the perspective of the Pythagorean theorem and further points out that the PCA can be extended to more general loss function—“robust" or “sparse" PCA; the second view reveals the relationship between tensor decomposition and PCA, which leads to a new idea of constructing tensor decomposition from the PCA perspective; the last view shows that higher-order PCA has a limitation to reveal anisotropic structure and further points out that a new method, “depth PCA," can be used to conquer this limitation.

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