Tensor Valued Common and Individual Feature Extraction: Multi-dimensional Perspective

Modern datasets in data science applications have immense volume, veracity, velocity and variety (the for V's of big data) [1, 2], and often exhibit a large degree of structural richness among their entries. These data characteristics are often prohibitive to the application of classical matrix algebra as its "flat-view" way of operation cannot cope with the sheer volume of data and the corresponding imbalanced matrix structures, such as as "tall and narrow" or "short and wide" ones. On the other hand, when arranged in multidimensional structures (tensors), the same data often admit much more convenient and mathematically tractable ways of analysis, by virtue of the associated multi-linear algebra. However, until recently, such an approach to data analysis was not very popular, due to high demand for storage and computational resources. There are several ways to tensorize data prior to further analysis, such as through: (i) natural tensor formation, (ii) experimental design, or (iii) mathematical construction [3]. This flexibility and a highly informative nature of multi-way data representation is supported by 1 Figure 1: Efficient representation of an imbalanced block-matrix structure (a set of video frames, top row) in the form of much more convenient and flexible tensor structure (a cube of frames, bottom row).