Tensor Decomposition

Tensor decomposition (TD) is a natural extension of matrix factorization (MF), introduced for matrices in the previous chapter, when tensors instead of matrices are considered. In contrast to the MF that is usually represented as a product of two matrices, TD has various forms. In contrast to the matrices that were extensively studied over long period, tensor has much shorter history of extensive investigations, especially from the application point of views. Thus, there are no de facto standards to be used for the specific application. Similar to the aim of MF, that of TD is also to reduce the degrees of freedoms. Nevertheless, how the degrees of freedom can be reduced has many variations for TD. In this chapter, we introduce three principal realizations of TD: sum of outer product of vectors, product summation of (smaller) tensor and matrices, and product summation of (smaller) tensors. These three methods have their own unique pros and cons. In addition to the algorithm to perform each of TDs, we will also discuss about these pros and cons of three methods introduced.