Calculating the Projective Norm of Higher-Order Tensors Using a Gradient Descent Algorithm

Projective Norms are a class of tensor norms that map from the input to the 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 exhibits stable convergence to a minimum nuclear-rank decomposition of the given tensor in all our numerical experiments. We further extend the algorithm to symmetric tensors and to 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 supporting these results.