This work introduces E3x, a software package for building neural networks
that are equivariant with respect to the Euclidean group $\mathrm{E}(3)$,
consisting of translations, rotations, and reflections of three-dimensional
space. Compared to ordinary neural networks, $\mathrm{E}(3)$-equivariant models
promise benefits whenever input and/or output data are quantities associated
with three-dimensional objects. This is because the numeric values of such
quantities (e.g. positions) typically depend on the chosen coordinate system.
Under transformations of the reference frame, the values change predictably,
but the underlying rules can be difficult to learn for ordinary machine
learning models. With built-in $\mathrm{E}(3)$-equivariance, neural networks
are guaranteed to satisfy the relevant transformation rules exactly, resulting
in superior data efficiency and accuracy. The code for E3x is available from
this https URL

这项工作介绍了 E3x，这是一个用于构建神经网络的软件包，它对于欧几里得群 E (3) 是等变的，包括三维空间的平移、旋转和反射。与普通的神经网络相比，E (3) 等变模型在输入和 / 或输出数据为与三维对象相关的量时具有优势。因为这些量的数值值（例如位置）通常取决于所选的坐标系。在参考坐标系的变换下，值会发生可预测的变化，但对于普通的机器学习模型来说，学习这些基本规则可能会很困难。内置 E (3) 等变性后，神经网络保证完全符合相关的变换规则，从而具有卓越的数据效率和准确性。E3x 的代码可以从此 https URL 获得。

E3x: E (3)- 等变深度学习简易化

E3x: $\mathrm{E}(3)$-Equivariant Deep Learning Made Easy

The big empirical success of group equivariant networks has led in recent
years to the sprouting of a great variety of equivariant network architectures.
A particular focus has thereby been on rotation and reflection equivariant CNNs
for planar images. Here we give a general description of $E(2)$-equivariant
convolutions in the framework of Steerable CNNs. The theory of Steerable CNNs
thereby yields constraints on the convolution kernels which depend on group
representations describing the transformation laws of feature spaces. We show
that these constraints for arbitrary group representations can be reduced to
constraints under irreducible representations. A general solution of the kernel
space constraint is given for arbitrary representations of the Euclidean group
$E(2)$ and its subgroups. We implement a wide range of previously proposed and
entirely new equivariant network architectures and extensively compare their
performances. $E(2)$-steerable convolutions are further shown to yield
remarkable gains on CIFAR-10, CIFAR-100 and STL-10 when used as a drop-in
replacement for non-equivariant convolutions.

该研究描述了在可旋转卷积神经网络框架中的 $E (2)$- 等变卷积，提出了转换特性表示描述特征空间变换法则的群表示。研究人员证明了这些约束可以通过使用不可约表示约简为任意群表示的约束，并通过实现一系列先前提出的和全新的等变网络架构进行了广泛比较，表明当用作非等变卷积的替代品时，在 CIFAR-10、CIFAR-100 和 STL-10 上使用 $E (2)$- 可旋转卷积可以取得显著的改进。

广义 E（2）等变可转 CNN

General $E(2)$-Equivariant Steerable CNNs

Many geometric estimation problems take the form of synchronization over the
special Euclidean group: estimate the values of a set of poses given noisy
measurements of a subset of their pairwise relative transforms. This problem is
typically formulated as a maximum-likelihood estimation that requires solving a
nonconvex nonlinear program, which is computationally intractable in general.
Nevertheless, in this paper we present an algorithm that is able to efficiently
recover certifiably globally optimal solutions of this estimation problem in a
non-adversarial noise regime. The crux of our approach is the development of a
semidefinite relaxation of the maximum-likelihood estimation whose minimizer
provides the exact MLE so long as the magnitude of the noise corrupting the
available measurements falls below a certain critical threshold; furthermore,
whenever exactness obtains, it is possible to verify this fact a posteriori,
thereby certifying the optimality of the recovered estimate. We develop a
specialized optimization scheme for solving large-scale instances of this
semidefinite relaxation by exploiting its low-rank, geometric, and
graph-theoretic structure to reduce it to an equivalent optimization problem on
a low-dimensional Riemannian manifold, and then design a Riemannian
truncated-Newton trust-region method to solve this reduction efficiently. We
combine this fast optimization approach with a simple rounding procedure to
produce our algorithm, SE-Sync. Experimental evaluation on a variety of
simulated and real-world pose-graph SLAM datasets shows that SE-Sync is capable
of recovering globally optimal solutions when the available measurements are
corrupted by noise up to an order of magnitude greater than that typically
encountered in robotics applications, and does so at a computational cost that
scales comparably with that of direct Newton-type local search techniques.

本文介绍一种解决几何估计中同步问题的算法，通过基于半定松弛的最大似然估计得出全局最优解，该算法可以在非对抗噪声情况下高效恢复可认证的全局最优解，并在大规模实例上使用低维黎曼曲面与特殊优化方案实现了优化问题的求解。