Optimal Evaluation of Symmetry-Adapted $n$-Correlations Via Recursive Contraction of Sparse Symmetric Tensors

TL;DR

This paper introduces an optimal recursive algorithm for evaluating high-dimensional, symmetry-invariant polynomials by contracting sparse symmetric tensors, significantly improving computational efficiency for large degrees.

Contribution

It provides a novel recursive contraction method for symmetric tensors that is proven to be optimal as polynomial degree approaches infinity.

Findings

The proposed algorithm is optimal in the limit of infinite polynomial degree.

It effectively handles high-dimensional, permutation- and rotation-invariant polynomials.

The method reduces computational complexity in evaluating symmetry-adapted correlations.

Abstract

We present a comprehensive analysis of an algorithm for evaluating high-dimensional polynomials that are invariant under permutations and rotations. The key bottleneck is the contraction of a high-dimensional symmetric and sparse tensor with a specific sparsity pattern that is directly related to the symmetries imposed on the polynomial. We propose an explicit construction of a recursive evaluation strategy and show that it is optimal in the limit of infinite polynomial degree.