On calculation of canonical decomposition of Tensor via the grid of local discrepancies

TL;DR

This paper introduces a computationally efficient method for calculating the canonical decomposition of high-order tensors by replacing the global error norm with local discrepancies on hyperplanes, optimized via Newton and Monte Carlo methods.

Contribution

It proposes a novel approach that uses local discrepancies and Monte Carlo estimation to compute tensor canonical decompositions more efficiently.

Findings

Numerical tests confirm the method's efficiency for sixth order tensors.

The approach reduces computational resources compared to traditional methods.

The method accurately approximates high-order tensors using local discrepancy minimization.

Abstract

The method for calculation of the canonical decomposition that approximates a tensor of high order is considered, which requires moderate computational resources. It is based on the replacement of the approximation error norm (global discrepancy functional) by the grid of local functionals (discrepancies computed on hyperplanes). The point of the global functional minimum in the space of the canonical decomposition cores is determined by the set of the stationary points of local functionals. In result, the estimation of the cores of the canonical decomposition is possible using Newton method applied point-wisely along coordinates and nodes. The discrepancies on the hyperplanes are calculated using Monte-Carlo method. Numerical tests on the approximation of sixth order tensors confirms the efficiency of the proposed approach.