A Levenberg-Marquardt Method for Tensor Approximation

Abstract

This paper presents a tensor approximation algorithm, based on the Levenberg–Marquardt method for the nonlinear least square problem, to decompose large-scale tensors into the sum of the products of vector groups of a given scale, or to obtain a low-rank tensor approximation without losing too much accuracy. An Armijo-like rule of inexact line search is also introduced into this algorithm. The result of the tensor decomposition is adjustable, which implies that the decomposition can be specified according to the users’ requirements. The convergence is proved, and numerical experiments show that it has some advantages over the classical Levenberg–Marquardt method. This algorithm is applicable to both symmetric and asymmetric tensors, and it is expected to play a role in the field of large-scale data compression storage and large-scale tensor approximation operations.

Keywords

• tensor decomposition
• Levenberg–Marquardt method
• non-linear least squares problem
• damped method