This paper proposes an adjoint-equivariant neural network that takes Lie
algebra data as input. Various types of equivariant neural networks have been
proposed in the literature, which treat the input data as elements in a vector
space carrying certain types of transformations. In comparison, we aim to
process inputs that are transformations between vector spaces. The change of
basis on transformation is described by conjugations, inducing the
adjoint-equivariance relationship that our model is designed to capture.
Leveraging the invariance property of the Killing form, the proposed network is
a general framework that works for arbitrary semisimple Lie algebras. Our
network possesses a simple structure that can be viewed as a Lie algebraic
generalization of a multi-layer perceptron (MLP). This work extends the
application of equivariant feature learning. As an example, we showcase its
value in homography modeling using sl(3) Lie algebra.

本文提出了一个以李代数数据作为输入的伴随等变性神经网络，通过共轭描述变换基的变化，利用杀正则化的不变性特性，该网络是一个可适用于任意半单李代数的简单结构，可作为多层感知器的李代数推广，扩展了等变性特征学习的应用场景，并展示了在使用 sl (3) 李代数进行同态建模中的价值。

李代数 Lie 神经网络：伴随等变型的半单李代数神经网络

Lie Neurons: Adjoint-Equivariant Neural Networks for Semisimple Lie  Algebras

This paper proposes a correspondence-free method for point cloud rotational
registration. We learn an embedding for each point cloud in a feature space
that preserves the SO(3)-equivariance property, enabled by recent developments
in equivariant neural networks. The proposed shape registration method achieves
three major advantages through combining equivariant feature learning with
implicit shape models. First, the necessity of data association is removed
because of the permutation-invariant property in network architectures similar
to PointNet. Second, the registration in feature space can be solved in
closed-form using Horn's method due to the SO(3)-equivariance property. Third,
the registration is robust to noise in the point cloud because of the joint
training of registration and implicit shape reconstruction. The experimental
results show superior performance compared with existing correspondence-free
deep registration methods.

本文提出了一种基于等变特征学习和隐式形状模型的无对应点云旋转配准方法，可实现无需数据关联、在特征空间中闭式求解和对噪声鲁棒。实验结果表明，该方法优于现有基于深度学习的无对应点云配准方法。