Tensor Train Decomposition

Abstract

Transmit Train (TT)-format is simple but powerful tensor decomposition and most widely used in many of the fields faced with the curse of dimensions. CANDECOMP / PARAFAC (CP) order-3 algorithms, also called canonical polyadic decomposition (CPD), are easy to implement and can be generalised to CPD of higher order. Unfortunately, the algorithms are becoming computationally challenging and are mostly not acceptable to tensors of higher order and fairly large scale. If the canonical rank R is small thus the parameters of the tensor will depend on N-dimensions linearly so there many attempts to find out the best low-rank approximation. The TT decomposition has numerous applications in several areas such as chemo-metrics, linear algebra, numerical analysis, computer vision, psycho-metrics are also discussed. In this thesis, we investigate the background of tensor required for the tensor-train format, best low-rank approximation algorithms, and couple of its important applications.