Calculating the Projective Norm of higher-order tensors using a gradient descent algorithm

TL;DR

This paper introduces a gradient descent algorithm to efficiently estimate the projective norm of higher-order tensors, addressing the NP-hard challenge and extending to symmetric tensors and density matrices.

Contribution

A novel gradient descent method for computing the projective norm of higher-order tensors, with guarantees of convergence and extensions to symmetric tensors and density matrices.

Findings

Successfully computes nuclear rank and projective norm for pure and mixed states.

Demonstrates convergence and effectiveness through numerical experiments.

Extends the algorithm to symmetric tensors and density matrices.

Abstract

Projective Norms are a class of tensor norms that map on the input and output spaces. These norms are useful for providing a measure of entanglement. Calculating the projective norms is an NP-hard problem, which creates challenges in computing due to the complexity of the exponentially growing parameter space for higher-order tensors. We develop a novel gradient descent algorithm to estimate the projective norm of higher-order tensors. The algorithm guarantees convergence to a minimum nuclear rank decomposition of our given tensor. We extend our algorithm to symmetric tensors and density matrices. We demonstrate the performance of our algorithm by computing the nuclear rank and the projective norm for both pure and mixed states and provide numerical evidence for the same.