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Constructing 3D Rotational Invariance and Equivariance with Symmetric Tensor Networks

Meng Zhang, Chao Wang, Hao Zhang, Shaojun Dong, Lixin He

TL;DR

The paper delivers a unified, constructive framework for fully characterizing continuous $SO(3)$-invariant and -equivariant functions when inputs and outputs are tuples of Cartesian or spherical tensors. It introduces tensor network generators built from isotropic building blocks ($oldsymbol\\delta$, $oldsymbol\\epsilon$, and $P_l$) and shows how invariant functions can be composed with ordinary neural networks, while equivariant maps arise by differentiating these invariants. The framework provides explicit, finite forms for common vector-input/either Cartesian or spherical tensor-output mappings and demonstrates that many established GNN primitives are special cases within this construction. It also extends naturally to $O(3)$ via parity considerations and offers practical diagrammatic tools for symmetry-preserving design in geometric GNNs. The approach promises data-efficient, physics-informed learning by embedding rotational symmetry directly into the model architecture.

Abstract

Symmetry-aware architectures are central to geometric deep learning. We present a systematic approach for constructing continuous rotationally invariant and equivariant functions using symmetric tensor networks. The proposed framework supports inputs and outputs given as a tuple of Cartesian tensors of different rank as well as spherical tensors of different type. We introduce tensor network generators for invariant maps and obtain equivariant maps via differentiation. Specifically, we derive general continuous equivariant maps from vector inputs to Cartesian or spherical tensor output. Finally, we clarify how common equivariant primitives in geometric graph neural networks arise within our construction.

Constructing 3D Rotational Invariance and Equivariance with Symmetric Tensor Networks

TL;DR

The paper delivers a unified, constructive framework for fully characterizing continuous -invariant and -equivariant functions when inputs and outputs are tuples of Cartesian or spherical tensors. It introduces tensor network generators built from isotropic building blocks (, , and ) and shows how invariant functions can be composed with ordinary neural networks, while equivariant maps arise by differentiating these invariants. The framework provides explicit, finite forms for common vector-input/either Cartesian or spherical tensor-output mappings and demonstrates that many established GNN primitives are special cases within this construction. It also extends naturally to via parity considerations and offers practical diagrammatic tools for symmetry-preserving design in geometric GNNs. The approach promises data-efficient, physics-informed learning by embedding rotational symmetry directly into the model architecture.

Abstract

Symmetry-aware architectures are central to geometric deep learning. We present a systematic approach for constructing continuous rotationally invariant and equivariant functions using symmetric tensor networks. The proposed framework supports inputs and outputs given as a tuple of Cartesian tensors of different rank as well as spherical tensors of different type. We introduce tensor network generators for invariant maps and obtain equivariant maps via differentiation. Specifically, we derive general continuous equivariant maps from vector inputs to Cartesian or spherical tensor output. Finally, we clarify how common equivariant primitives in geometric graph neural networks arise within our construction.

Paper Structure

This paper contains 28 sections, 12 theorems, 63 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Tensor network
  4. Group invariance and equivariance
  5. Symmetric tensor network
  6. Cartesian and spherical tensors
  7. Tensor network generators
  8. Vector inputs
  9. Cartesian tensor inputs
  10. Spherical tensor inputs
  11. Constructing continuous invariant functions
  12. Generalization to equivariant functions
  13. Equivariant functions with vector inputs
  14. Constructing equivariant operations for geometric graph neural network
  15. Constructing equivariant operations for Cartesian tensor feature
  16. ...and 13 more sections

Key Result

Lemma 3.1

Let inputs $\bm x_1,\dots,\bm x_n \in \mathbb{R}^{3}$. The set of generators of $\mathbb R[V]^{SO(3)}$ is $\{\bm x_i\cdot \bm x_j,(\bm x_i\times \bm x_j)\cdot \bm x_k\}$. Therefore any invariant polynomial takes the form of $f(\bm x)=q(\{ \bm x_i\cdot \bm x_j,(\bm x_i\times \bm x_j )\cdot \bm x_k \}

Figures (4)

  • Figure 1: (a) Graphical representation of a one-leg tensor (vector) and a two-leg tensor (matrix). (b) The contraction of tensor $A$ and $B$, this is the matrix multiplication $C_{ik} = \sum_j A_{ij}B_{jk}$. (c) The derivative of a tensor network with respect to a specific tensor $T$, the result of which is a tensor network where tensor $T$ is removed. (d) Tensor train decomposition, namely, a N-leg tensor is decomposed to 2-leg and 3-leg tensors.
  • Figure 2: Symmetric tensors and symmetric tensor networks. (a) The graphical illustration of the Equation (\ref{['eqn:sym_cond']}). (b) Tensor networks that consist of symmetric tensors are also symmetric tensors as a whole. We first insert identity $\rho_i(g) \rho_i(g)^{T} = I$ on the contracted leg. Since every tensor in the network is symmetric, the tensor networks as a whole are also symmetric.
  • Figure 3: (a) The general form of continuous $SO(3)$ invariant functions $f_{\rm inv}$ by composing a general neural network $p$ with $g_1,\dots,g_k$, which are tensor network generators. (b) The general form of $SO(3)$ equivariant functions by composing a general neural network $p$ with $g_1,\dots,g_k$, which are tensor network generators, and then applying $T_{\rm up}$.
  • Figure 4: $T_{\rm up}$ transforms an invariant function into an equivariant function. $T_{\rm down}$ transforms an equivariant function into an invariant function.

Theorems & Definitions (23)

  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Lemma 3.1
  • Lemma 3.2
  • Proposition 3.3
  • Lemma 3.4
  • Proposition 3.5
  • Proposition 3.6
  • Lemma 4.1
  • ...and 13 more