Gramian Tensor Decomposition via Semidefinite Programming

TL;DR

This paper investigates the Gramian tensor decomposition problem, formulating it as a rank minimization task, and explores semidefinite programming relaxations to identify conditions for optimal solutions.

Contribution

It introduces a relaxation approach for Gramian tensor decomposition and analyzes when the relaxation yields solutions equivalent to the original problem.

Findings

Relaxation often yields optimal solutions in certain tensor rank and order cases.

Identifies instances where the relaxation fails to produce the original problem's solution.

Provides examples illustrating the complexity of tensor rank minimization.

Abstract

In this paper we examine a symmetric tensor decomposition problem, the Gramian decomposition, posed as a rank minimization problem. We study the relaxation of the problem and consider cases when the relaxed solution is a solution to the original problem. In some instances of tensor rank and order, we prove generically that the solution to the relaxation will be optimal in the original. In other cases, we present interesting examples and approaches that demonstrate the intricacy of this problem.