Emerging from low-level vision theory, steerable filters found their
counterpart in prior work on steerable convolutional neural networks
equivariant to rigid transformations. In our work, we propose a steerable
feed-forward learning-based approach that consists of neurons with spherical
decision surfaces and operates on point clouds. Such spherical neurons are
obtained by conformal embedding of Euclidean space and have recently been
revisited in the context of learning representations of point sets. Focusing on
3D geometry, we exploit the isometry property of spherical neurons and derive a
3D steerability constraint. After training spherical neurons to classify point
clouds in a canonical orientation, we use a tetrahedron basis to quadruplicate
the neurons and construct rotation-equivariant spherical filter banks. We then
apply the derived constraint to interpolate the filter bank outputs and, thus,
obtain a rotation-invariant network. Finally, we use a synthetic point set and
real-world 3D skeleton data to verify our theoretical findings. The code is
available at this https URL

该论文提出了一种可定向前馈的学习方法，利用球面神经元对点云进行分类，得到具有旋转等变性的滤波器组建网络，利用三角形基础来推导滤波器并构建旋转等变性的球形滤波器组。

可操纵的 3D 球形神经元

Steerable 3D Spherical Neurons

The effectiveness of Convolutional Neural Networks (CNNs) has been
substantially attributed to their built-in property of translation
equivariance. However, CNNs do not have embedded mechanisms to handle other
types of transformations. In this work, we pay attention to scale changes,
which regularly appear in various tasks due to the changing distances between
the objects and the camera. First, we introduce the general theory for building
scale-equivariant convolutional networks with steerable filters. We develop
scale-convolution and generalize other common blocks to be scale-equivariant.
We demonstrate the computational efficiency and numerical stability of the
proposed method. We compare the proposed models to the previously developed
methods for scale equivariance and local scale invariance. We demonstrate
state-of-the-art results on MNIST-scale dataset and on STL-10 dataset in the
supervised learning setting.

介绍了一种使用可控滤波器建立尺度等变卷积网络的通用理论，并将其他常见块推广为尺度等变块，此方法具有计算效率和数值稳定性，同时在 MNIST-scale 数据集和 STL-10 数据集中实现了领先水平的分类结果。

尺度等变可转移网络

Scale-Equivariant Steerable Networks

In many machine learning tasks it is desirable that a model's prediction
transforms in an equivariant way under transformations of its input.
Convolutional neural networks (CNNs) implement translational equivariance by
construction; for other transformations, however, they are compelled to learn
the proper mapping. In this work, we develop Steerable Filter CNNs (SFCNNs)
which achieve joint equivariance under translations and rotations by design.
The proposed architecture employs steerable filters to efficiently compute
orientation dependent responses for many orientations without suffering
interpolation artifacts from filter rotation. We utilize group convolutions
which guarantee an equivariant mapping. In addition, we generalize He's weight
initialization scheme to filters which are defined as a linear combination of a
system of atomic filters. Numerical experiments show a substantial enhancement
of the sample complexity with a growing number of sampled filter orientations
and confirm that the network generalizes learned patterns over orientations.
The proposed approach achieves state-of-the-art on the rotated MNIST benchmark
and on the ISBI 2012 2D EM segmentation challenge.

本文介绍了一种基于可平移、可旋转等变性的创新神经网络结构，采用可旋转滤波器和群卷积实现了对变换的兼容性，使用改进的权重初始化方案来提高准确率，在旋转 MNIST 基准测试和 ISBI 2012 2D EM 语义分割挑战中取得了良好表现。