Features that are equivariant to a larger group of symmetries have been shown
to be more discriminative and powerful in recent studies. However, higher-order
equivariant features often come with an exponentially-growing computational
cost. Furthermore, it remains relatively less explored how rotation-equivariant
features can be leveraged to tackle 3D shape alignment tasks. While many past
approaches have been based on either non-equivariant or invariant descriptors
to align 3D shapes, we argue that such tasks may benefit greatly from an
equivariant framework. In this paper, we propose an effective and practical
SE(3) (3D translation and rotation) equivariant network for point cloud
analysis that addresses both problems. First, we present SE(3) separable point
convolution, a novel framework that breaks down the 6D convolution into two
separable convolutional operators alternatively performed in the 3D Euclidean
and SO(3) spaces. This significantly reduces the computational cost without
compromising the performance. Second, we introduce an attention layer to
effectively harness the expressiveness of the equivariant features. While
jointly trained with the network, the attention layer implicitly derives the
intrinsic local frame in the feature space and generates attention vectors that
can be integrated into different alignment tasks. We evaluate our approach
through extensive studies and visual interpretations. The empirical results
demonstrate that our proposed model outperforms strong baselines in a variety
of benchmarks

本文提出了一个有效的 SE（3）网络，它是针对点云分析领域中 3D 形状对齐任务而设计的，可以利用等变特征来提高性能并解决相对较少探索的旋转等变特征对 3D 形状对齐任务的处理问题。我们采用一个新的框架 SE (3) 可分点卷积来降低计算成本，并在网络中引入一个注意力层来有效地利用等变特征的表达能力。通过广泛的研究和视觉解释，实证结果表明，我们提出的模型在各种基准测试中优于强基线。

用于三维点云分析的等变点网络

Equivariant Point Network for 3D Point Cloud Analysis

This paper is concerned with a fundamental problem in geometric deep learning
that arises in the construction of convolutional neural networks on surfaces.
Due to curvature, the transport of filter kernels on surfaces results in a
rotational ambiguity, which prevents a uniform alignment of these kernels on
the surface. We propose a network architecture for surfaces that consists of
vector-valued, rotation-equivariant features. The equivariance property makes
it possible to locally align features, which were computed in arbitrary
coordinate systems, when aggregating features in a convolution layer. The
resulting network is agnostic to the choices of coordinate systems for the
tangent spaces on the surface. We implement our approach for triangle meshes.
Based on circular harmonic functions, we introduce convolution filters for
meshes that are rotation-equivariant at the discrete level. We evaluate the
resulting networks on shape correspondence and shape classifications tasks and
compare their performance to other approaches.

该论文提出了一种用于表面的网络架构，其中包含旋转等变的特征，以解决运输滤波器核时出现的旋转歧义问题。基于圆谐函数的卷积滤波器在离散层次上具有旋转等变性，将该方法应用于三角网中，并在形状对应和分类任务上进行了评估。

使用旋转等变特征的曲面上的 CNN

CNNs on Surfaces using Rotation-Equivariant Features

We propose a novel quaternion product unit (QPU) to represent data on 3D
rotation groups. The QPU leverages quaternion algebra and the law of 3D
rotation group, representing 3D rotation data as quaternions and merging them
via a weighted chain of Hamilton products. We prove that the representations
derived by the proposed QPU can be disentangled into "rotation-invariant"
features and "rotation-equivariant" features, respectively, which supports the
rationality and the efficiency of the QPU in theory. We design quaternion
neural networks based on our QPUs and make our models compatible with existing
deep learning models. Experiments on both synthetic and real-world data show
that the proposed QPU is beneficial for the learning tasks requiring rotation
robustness.

提出一种新型四元数乘积单元（QPU），利用四元数代数和三维旋转群规律表示三维旋转数据，并通过一条带权重的 Hamilton 乘积链合并它们。通过实验证明，基于该 QPU 的四元数神经网络在需要旋转稳定性的学习任务中有优势，并且能够得到旋转不变特征和旋转等变特征的优秀表现，从而确保其理论上的合理性和高效性。