On Multilinear Principal Component Analysis of Order-Two Tensors

TL;DR

This paper develops the asymptotic theory for Multilinear Principal Component Analysis (MPCA) of order-two tensors, demonstrating its advantages over traditional PCA in reducing dimensionality while preserving data structure, with applications to face image data.

Contribution

It provides the first asymptotic distribution theory for order-two MPCA, explaining its statistical benefits and showing improved face reconstruction compared to PCA.

Findings

MPCA has better dimensionality reduction efficiency than PCA.

Asymptotic distributions for MPCA components are derived.

MPCA improves face reconstruction in Olivetti Faces dataset.

Abstract

Principal Component Analysis (PCA) is a commonly used tool for dimension reduction in analyzing high dimensional data; Multilinear Principal Component Analysis (MPCA) has the potential to serve the similar function for analyzing tensor structure data. MPCA and other tensor decomposition methods have been proved effective to reduce the dimensions for both real data analyses and simulation studies (Ye, 2005; Lu, Plataniotis and Venetsanopoulos, 2008; Kolda and Bader, 2009; Li, Kim and Altman, 2010). In this paper, we investigate MPCA's statistical properties and provide explanations for its advantages. Conventional PCA, vectorizing the tensor data, may lead to inefficient and unstable prediction due to its extremely large dimensionality. On the other hand, MPCA, trying to preserve the data structure, searches for low-dimensional multilinear projections and decreases the dimensionality efficiently. The asymptotic theories for order-two MPCA, including asymptotic distributions for principal components, associated projections and the explained variance, are developed. Finally, MPCA is shown to improve conventional PCA on analyzing the {\sf Olivetti Faces} data set, by constructing more module oriented basis in reconstructing the test faces.