Tensor Approximation in Visualization and Graphics: Background Theory

This compendium on tensor approximation (TA) gives an overview on typical tensor approximation notation and definitions. TA is a tool for data approximation in higher orders. Precisely speaking, TA is an higher-order extension of the matrix singular value decomposition and is a generalization of a data factorization of multidimensional datasets into a set of bases and coefficients. TA consists of two main parts: the tensor decomposition and the tensor reconstruction. In TA, there are several decomposition models available, which are summarized in this document including the main different decomposition algorithms. Furthermore, since low-rank tensor approximations is an interesting tool for data reduction and data factorization, the tensor rank reduction is another important topic. For interactive visualization and graphics applications, the tensor reconstruction is another critical issue since often a fast real-time reconstruction process is required. In this compendium, several reconstruction processes for the different TA models are presented. Finally, some particular TA bases properties that are useful for computer graphics or scientific visualization applications are outlined.