Real-world geometry and 3D vision tasks are replete with challenging
symmetries that defy tractable analytical expression. In this paper, we
introduce Neural Isometries, an autoencoder framework which learns to map the
observation space to a general-purpose latent space wherein encodings are
related by isometries whenever their corresponding observations are
geometrically related in world space. Specifically, we regularize the latent
space such that maps between encodings preserve a learned inner product and
commute with a learned functional operator, in the same manner as rigid-body
transformations commute with the Laplacian. This approach forms an effective
backbone for self-supervised representation learning, and we demonstrate that a
simple off-the-shelf equivariant network operating in the pre-trained latent
space can achieve results on par with meticulously-engineered, handcrafted
networks designed to handle complex, nonlinear symmetries. Furthermore,
isometric maps capture information about the respective transformations in
world space, and we show that this allows us to regress camera poses directly
from the coefficients of the maps between encodings of adjacent views of a
scene.

实际几何和 3D 视觉任务充满了令人困惑的对称性，该论文介绍了一种名为神经等距映射的自动编码器框架，它学习将观察空间映射到通用的潜在空间，在这个空间中，当相应的观察在世界空间中有几何关系时，编码是由等距映射相关的。这种方法形成了一个有效的自监督表示学习的基础，并且我们证明了一个在预训练的潜在空间中操作的简单暗箱神经网络能够达到与精心设计的手工网络相媲美的结果，以处理复杂的非线性对称性。此外，等距映射捕捉了世界空间中相应变换的信息，我们展示了这使我们能够直接从相邻视图的编码之间的映射的系数为相机姿态回归。

神经等度：驯服等变机器学习的变换

Neural Isometries: Taming Transformations for Equivariant ML

The usage of 3D vision algorithms, such as shape reconstruction, remains
limited because they require inputs to be at a fixed canonical rotation.
Recently, a simple equivariant network, Vector Neuron (VN) has been proposed
that can be easily used with the state-of-the-art 3D neural network (NN)
architectures. However, its performance is limited because it is designed to
use only three-dimensional features, which is insufficient to capture the
details present in 3D data. In this paper, we introduce an equivariant feature
representation for mapping a 3D point to a high-dimensional feature space. Our
feature can discern multiple frequencies present in 3D data, which is the key
to designing an expressive feature for 3D vision tasks. Our representation can
be used as an input to VNs, and the results demonstrate that with our feature
representation, VN captures more details, overcoming the limitation raised in
its original paper.

介绍了一种用于将 3D 点映射到高维特征空间的等变特征表示方法，该特征能够识别 3D 数据中存在的多个频率，以提供更多细节来克服原始研究中的限制。