Algorithms for Tensor Decomposition

Authors: Enrica Barrilli, Oriol Reig

Tensor decomposition is an important problem in the study of tensor properties, since it allows to reveal hidden structures. Several types of decomposition and ranks are investigated:

• Rank or Waring Decomposition
• Generalized Additive Decomposition
• Cactus-Rank Decomposition
• Border rank

Tensor Low Rank Approximation problems are also studied.

Tensor decompositions

Rank or Waring decomposition

This decomposition also called rank decomposition consists in decomposing a symmetric tensor $F$ of order $d$ as

$$F=\sum _{i=1}^r{\omega }_i{L}_i^d$$

where ${\omega }_i\in \mathbb{k}$, $L_i$ is a linear form (of degree $1$). The minimal $r$ is called the rank of $F$.

Generalized Additive Decomposition

A Generalized Additive Decomposition (GAD) of a symmetric tensor $F$ of order $d$ is

$$F=\sum _{i=1}^r{\omega }_i{L}_i^{d-k_i}$$

where ${\omega }_i$ is a symmetric tensor of degree $k_i$, $L_i$ is a linear form (of degree $1$).

Cactus rank decomposition

Border rank decomposition

Algorithms

Courses

• Symmetric Tensors and Their Decomposition: From Basics to Sylvester’s Algorithm
  Alessandra Bernardi (UniTN) Slides Recording (Part 1) Recording (Part 2)
• Algebra for the analysis of tensors and moment sequences Bernard Mourrain (Inria) Slides Recording (Part 1) Recording (Part 2)
• The cheapest tensors (tensors of minimal border rank)|Joseph Landsberg (Texas A&M University) Slides Part 1 Slides Part 2 Exercises Recording (Part 1) Recording| (Part 2)