Visualization of the Eigen-structure Updating of Three-Dimensional Random Tensor

Abstract

This image visualizes the structural relationship between the higher order singular value decomposition (HOSVD) of a three-dimensional random tensor (above) of size 20x15x10 and a partial three-dimensional tensor (below) of size 7x10x5, which is extracted from the original random tensor (above). The relative position of the new tensor within the old tensor is highlighted in yellow. Their higher order singular value decompositions are shown on the right, which consist of core tensors (red and blue), and two sets of loading matrices (shown in colored images). In this research in UIC, we discovered the structural relationship between the HOSVDs of those two tensors, and developed fast updating and downdating algorithms for re-calculating of HOSVDs. These algorithms are applicable to many areas of research involving tensor-form data. Specifically, we applied these algorithms in dynamic updating and downdating of video and motion trajectory databases.