Tensor Decompositions, State of the Art and Applications

TL;DR

This paper surveys tensor algebra tools used in statistics and signal processing, discusses challenges in extending linear algebra decompositions to tensors, and reviews numerical algorithms and their limitations.

Contribution

It provides a partial survey of tensor decompositions, highlighting difficulties in defining tensor rank and analyzing the limitations of existing algorithms.

Findings

Tensor decompositions are challenging to extend from linear algebra.

Calculating tensor rank is complex and problematic.

Numerical algorithms exist but have notable limitations.

Abstract

In this paper, we present a partial survey of the tools borrowed from tensor algebra, which have been utilized recently in Statistics and Signal Processing. It is shown why the decompositions well known in linear algebra can hardly be extended to tensors. The concept of rank is itself difficult to define, and its calculation raises difficulties. Numerical algorithms have nevertheless been developed, and some are reported here, but their limitations are emphasized. These reports hopefully open research perspectives for enterprising readers.