构建可支持旋转不变与等变的对称张量网络框架
Constructing 3D Rotational Invariance and Equivariance with Symmetric Tensor Networks
- 用对称张量网络系统构造旋转不变/等变函数
- 从向量输入生成各类张量输出的通用等变映射
- 统一解释图神经网络中的常见等变组件
对称性感知架构是几何深度学习的核心。本文提出一种系统方法，利用对称张量网络构建连续旋转不变与等变函数。该框架支持以不同阶数的笛卡尔张量或不同类型球张量作为输入和输出。我们引入张量网络生成器来构造不变映射，并通过微分获得等变映射。具体地，从向量输入推导出通用于笛卡尔或球张量输出的连续等变映射。最后，阐明了几何图神经网络中常见等变基元在本构造中的来源。
原文摘要 · Abstract (English)
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.
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