Tensor decomposition and its applications

Abstract

This dissertation reviews classical vector - tensor analysis, building up to the necessary techniques required to decompose a tensor into a tensor train and to reconstruct it back into the original tensor with minimal error. The tensor train decomposition decomposes a tensor of dimensionality d into a train of d third order tensors, whose sizes are dependent upon the rank and chosen error bound. I will be reviewing the required operations of matricization, tensor - matrix, vector and tensor multiplication to be able to compute this decomposition. I then move onto analysing the tensor train decomposition by ap-plying it to different types of tensor, of differing dimensionality with a variety of accuracy bounds to investigate their influence on the time taken to complete the decomposition and the final absolute error. Finally I explore a method to compute a d-dimensional integration from the tensor train, which will allow larger tensors to be integrated with the memory required dramatically reduced after the tensor is decomposed. I will be applying this technique to two tensors with different ranks and compare the efficiency and accuracy of integrating directly from the tensor to that of the tensor train decomposition.