# Efficient computation of higher order cumulant tensors

## TL;DR

This paper presents a new tensor-based algorithm for efficiently computing higher order cumulants of multidimensional data, significantly reducing computational complexity and memory usage compared to previous methods.

## Contribution

It introduces a novel, super-symmetry exploiting algorithm for arbitrary order cumulant tensors, improving efficiency over existing approaches.

## Key findings

- Reduces computational complexity by approximately d!
- Decreases memory requirements for cumulant calculation
- Applicable to high-dimensional, higher-order cumulant computation

## Abstract

In this paper, we introduce a novel algorithm for calculating arbitrary order cumulants of multidimensional data. Since the $d^\text{th}$ 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.