Error Analysis of TT-Format Tensor Algorithms

Abstract

The tensor train (TT) decomposition is a representation technique for arbitrary tensors, which allows efficient storage and computations. For a d-dimensional tensor with d ≥ 2, that decomposition consists of two ordinary matrices and d − 2 third-order tensors. In this paper we prove that the TT decomposition of an arbitrary tensor can be computed (or approximated, for data compression purposes) by means of a backward stable algorithm based on computations with Householder matrices. Moreover, multilinear forms with tensors represented in TT format can be computed efficiently with a small backward error.