  We propose a new algorithm for computing the tensor rank decomposition or
canonical polyadic decomposition of higher-order tensors subject to a rank and
genericity constraint. Reformulating this as a system of polynomial equations
allows us to leverage recent numerical linear algebra tools from computational
algebraic geometry. We describe the complexity of our algorithm in terms of the
multigraded regularity of a multihomogeneous ideal. We prove effective bounds
for many formats and ranks and conjecture a general formula. These bounds are
of independent interest for overconstrained polynomial system solving. Our
experiments show that our algorithm can outperform state-of-the-art algebraic
algorithms by an order of magnitude in terms of accuracy, computation time, and
memory consumption.
