Bounding Real Tensor Optimizations via the Numerical Range

TL;DR

This paper introduces a method using the numerical range of matrices to efficiently bound the optimal values of certain real tensor optimization problems, surpassing traditional eigenvalue bounds and semidefinite programming in speed.

Contribution

It presents a novel approach leveraging the numerical range to obtain tighter bounds for tensor optimization problems, with broad applications in linear algebra.

Findings

Stronger bounds than eigenvalue-based methods

Faster computation than semidefinite relaxations

Applications to matrix rank and positivity problems

Abstract

We show how the numerical range of a matrix can be used to bound the optimal value of certain optimization problems over real tensor product vectors. Our bound is stronger than the trivial bounds based on eigenvalues, and can be computed significantly faster than bounds provided by semidefinite programming relaxations. We discuss numerous applications to other hard linear algebra problems, such as showing that a real subspace of matrices contains no rank-one matrix, and showing that a linear map acting on matrices is positive.