# A new Algorithm for Overcomplete Tensor Decomposition based on Sums-of-Squares Optimisation

## TL;DR

This paper introduces a novel algorithm leveraging Sums-of-Squares Optimization for overcomplete tensor decomposition, enabling the extraction of individual components from high-degree polynomials through quadratic form reduction.

## Contribution

It develops a new class of algorithms based on Sums of Squares Programming to efficiently decompose overcomplete tensors by reducing polynomial degrees.

## Key findings

- Effective component extraction via eigenvalue decomposition.
- Reduction of high-degree polynomials to quadratic forms.
- Potential for improved tensor decomposition accuracy.

## Abstract

In this thesis, a new class of algorithms based on Sums of Squares Programming is developed. These allow to reduce a degree-$d$ homogeneous polynomial $T = \sum_{i = 1}^m \langle a_i, X \rangle^d $ to a quadratic form being close to a rank-$1$ form via a low-degree reduction polynomial $W\in\sum \mathbb{R}[X]^2$. $W$ can be thought of as a `weight function' attaining high values on merely one of the components $a_i$. The component can then be extracted by running an eigenvalue decomposition on the quadratic form $\sum_{i=1}^m W(a_i) \langle a_i, X \rangle^2$.