Efficient computation of higher order cumulant tensors

In this paper, we introduce a novel algorithm for calculating arbitrary order
cumulants of multidimensional data. Since the d'th order
cumulant can be presented in the form of an d-dimensional tensor, the
algorithm is presented using tensor operations. The algorithm provided in the
paper takes advantage of super-symmetry of cumulant and moment tensors.
We show that the proposed algorithm considerably reduces the computational
complexity and the computational memory requirement of cumulant calculation as
compared with existing algorithms. For the sizes of interest, the
reduction is of the order of d! compared to the naive algorithm.