Multilinear Principal Component Analysis (MPCA) for Tensor Data Dimensionality Reduction

Multilinear Principal Component Analysis (MPCA) is a dimensionality reduction technique specifically designed for higher-order tensor data, which effectively handles multidimensional data structures such as image sequences, videos, and 3D volumetric data. Unlike traditional Principal Component Analysis (PCA), MPCA performs decomposition directly in the tensor space rather than first flattening the data into vectors before dimensionality reduction, thereby better preserving the structural information of the data. MPCA was initially widely applied in facial recognition and gait recognition domains. Facial recognition data typically consists of 2D or 3D tensors (e.g., images under varying lighting conditions and poses), while gait recognition involves analyzing motion patterns across time sequences. MPCA can reduce computational complexity without losing critical features. With further research, this method has been extended to numerous application areas including medical image analysis, action recognition, and remote sensing data processing. The core concept of MPCA involves performing independent principal component analysis on each mode (such as row, column, and temporal dimensions), enabling dimensionality reduction while maintaining the higher-order structure of the data. This approach allows MPCA to more effectively capture multilinear relationships within the data, enhancing performance in classification or regression tasks. In MATLAB implementations, MPCA typically involves key operations including tensor unfolding and reconstruction, covariance matrix computation, and eigenvalue decomposition. Researchers can adjust parameters such as the number of retained principal components to optimize algorithm performance for different applications. The implementation often utilizes tensor manipulation functions (like tensor unfolding with 'reshape' operations) and eigenvalue decomposition functions ('eig' or 'svd') for mode-specific covariance matrices. Additionally, MPCA can be integrated with other machine learning methods such as Support Vector Machines (SVM) or deep learning models to further improve recognition accuracy through feature fusion techniques.