  Genuine multipartite entanglement of a given multipartite pure quantum state
can be quantified through its geometric measure of entanglement, which, up to
logarithms, is simply the maximum overlap of the corresponding unit tensor with
product unit tensors, a quantity that is also known as the injective norm of
the tensor. Our general goal in this work is to estimate this injective norm of
randomly sampled tensors. To this end, we study and compare various algorithms,
based either on the widely used alternating least squares method or on a novel
normalized gradient descent approach, and suited to either symmetrized or
non-symmetrized random tensors. We first benchmark their respective
performances on the case of symmetrized real Gaussian tensors, whose asymptotic
average injective norm is known analytically. Having established that our
proposed normalized gradient descent algorithm generally performs best, we then
use it to obtain numerical estimates for the average injective norm of complex
Gaussian tensors (i.e. up to normalization uniformly distributed multipartite
pure quantum states), with or without permutation-invariance. Finally, we also
estimate the average injective norm of random matrix product states constructed
from Gaussian local tensors, with or without translation-invariance. All these
results constitute the first numerical estimates on the amount of genuinely
multipartite entanglement typically present in various models of random
multipartite pure states.
