Flexible Moment-Invariant Bases from Irreducible Tensors

TL;DR

This work addresses degeneracies that hinder 3D moment-invariant bases, notably those arising from spherical functions, by marrying irreducible spherical-harmonic structure with Cartesian tensor contractions. It introduces irreducible tensor moment bases and proves that a basis of homogeneous invariants can be built from pure invariants of irreducible components plus a small fixed set of mixed invariants anchored via a robust tensor (Theorem t:pure2). Two practical descriptor-generation schemes are proposed: a Specific Flexible Basis with a central anchor and a Minimal Flexible Set that anchors all irreducibles pairwise, each providing complete, independent, and flexible invariants up to a chosen maximal order $\ell_m$. The Cartesian-spherical variant further specializes the theory for spherical functions, yielding more compact sets with comparable discriminative power. Applied to machine-learning interatomic potentials, the flexible descriptors improve accuracy (e.g., in HIP-HOP-NN for methane) and are implemented in open-source software, highlighting practical impact for robust, orientation-agnostic modeling in materials science.

Abstract

Moment invariants are a powerful tool for the generation of rotation-invariant descriptors needed for many applications in pattern detection, classification, and machine learning. A set of invariants is optimal if it is complete, independent, and robust against degeneracy in the input. In this paper, we show that the current state of the art for the generation of these bases of moment invariants, despite being robust against moment tensors being identically zero, is vulnerable to a degeneracy that is common in real-world applications, namely spherical functions. We show how to overcome this vulnerability by combining two popular moment invariant approaches: one based on spherical harmonics and one based on Cartesian tensor algebra.

Figures (12)

• Figure 1: The two functions from \ref{['3dScalarF']} are different but cannot be distinguished by Bujack's basis. $f_1$: pink, $f_2$: blue, with their magnitude depending on the angle visualized by the distance from the origin.
• Figure 2: The homogeneous third-order invariants in Bujack's basis in the same order as in Eq. \ref{['bujack']}.
• Figure 3: The homogeneous third-order invariants in Langbein's basis in the same order as in Eq. \ref{['langbein']}.
• Figure 4: A set of independent homogeneous third-order invariants in the basis formed by the irreducible tensor decomposition. The nodes all show order 3. The number of incident edges reveals their rank. The first four are the non-zero entries in the same order as in Eqs. \ref{['irreducibleTensor']} and \ref{['irreducibleTensor2']}. The last three involve the first rank irreducible tensor, which is zero in this case.
• Figure 5: Schematic illustration of the different flexible feature sets. The nodes are the irreducible tensors and the edges indicate the presence of a mixed invariant from a product of the two edges it connects. Three mixed invariants are typically needed to anchor the mutual alignment between two irreducible tensors.

Theorems & Definitions (28)

• Definition 1
• Lemma 1
• Lemma 2
• Example 1
• Example 2
• Definition 2
• Corollary 1
• Conjecture 1
• Example 3
• Lemma 3: Independent moments of an irreducible tensor