Multilinear principal component analysis (MPCA) is a multilinear extension of principal component analysis (PCA) that is used to analyze M-way arrays, also informally referred to as "data tensors".
[3] In 2000, De Lathauwer et al. restated Tucker and Kroonenberg's work in clear and concise numerical computational terms in their SIAM paper entitled "Multilinear Singular Value Decomposition",[4] (HOSVD) and in their paper "On the Best Rank-1 and Rank-(R1, R2, ..., RN ) Approximation of Higher-order Tensors".
The power of the tensor framework was showcased by analyzing human motion joint angles, facial images or textures in terms of their causal factors of data formation in the following works: Human Motion Signatures[6] (CVPR 2001, ICPR 2002), face recognition – TensorFaces,[7][8] (ECCV 2002, CVPR 2003, etc.)
Historically, MPCA has been referred to as "M-mode PCA", a terminology which was coined by Peter Kroonenberg in 1980.
[3] In 2005, Vasilescu and Terzopoulos introduced the Multilinear PCA[10] terminology as a way to better differentiate between linear and multilinear tensor decomposition, as well as, to better differentiate between the work[6][7][8][9] that computed 2nd order statistics associated with each data tensor mode(axis), and subsequent work on Multilinear Independent Component Analysis[10] that computed higher order statistics associated with each tensor mode/axis.