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Within statistics, Multilinear principal component analysis is a multilinear extension of principal component analysis. MPCA is employed in the analysis of n-way arrays, i.e. a cube or hyper-cube of numbers, also informally referred to as a "data tensor". N-way arrays may be decomposed, analyzed, or modeled by
The origin of MPCA can be traced back to the Tucker decomposition and Peter Kroonenberg's "M-mode PCA/3-mode PCA" work. 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", and in their paper "On the Best Rank-1 and Rank- Approximation of Higher-order Tensors".
Circa 2001, Vasilescu reframed the data analysis, recognition and synthesis problems as multilinear tensor problems based on the insight that most observed data are the compositional consequence of several causal factors of data formation, and are well suited for multi-modal data tensor analysis. 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, face recognition – TensorFaces, and computer graphics – TensorTextures.
Historically, MPCA has been referred to as "M-mode PCA", a terminology which was coined by Peter Kroonenberg in 1980. In 2005, Vasilescu and Terzopoulos introduced the Multilinear PCA terminology as a way to better differentiate between linear and multilinear tensor decomposition, as well as, to better differentiate between the work that computed 2nd order statistics associated with each data tensor mode, and subsequent work on Multilinear Independent Component Analysis that computed higher order statistics associated with each tensor mode/axis.