Tensor Decomposition

What is a 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

Low rank approximation problems are also studied.

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 $F$.

• 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)

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

• 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)