Generating point clouds, e.g., molecular structures, in arbitrary rotations,
translations, and enumerations remains a challenging task. Meanwhile, neural
networks utilizing symmetry invariant layers have been shown to be able to
optimize their training objective in a data-efficient way. In this spirit, we
present an architecture which allows to produce valid Euclidean distance
matrices, which by construction are already invariant under rotation and
translation of the described object. Motivated by the goal to generate
molecular structures in Cartesian space, we use this architecture to construct
a Wasserstein GAN utilizing a permutation invariant critic network. This makes
it possible to generate molecular structures in a one-shot fashion by producing
Euclidean distance matrices which have a three-dimensional embedding.

利用具有对称不变层的神经网络生成 Euclidean 距离矩阵，并使用其构建了一个 Wasserstein GAN，可以一次性生成具有三维嵌入的分子结构。

生成有效的欧几里得距离矩阵

Generating valid Euclidean distance matrices

Euclidean distance matrices (EDM) are matrices of squared distances between
points. The definition is deceivingly simple: thanks to their many useful
properties they have found applications in psychometrics, crystallography,
machine learning, wireless sensor networks, acoustics, and more. Despite the
usefulness of EDMs, they seem to be insufficiently known in the signal
processing community. Our goal is to rectify this mishap in a concise tutorial.
We review the fundamental properties of EDMs, such as rank or
(non)definiteness. We show how various EDM properties can be used to design
algorithms for completing and denoising distance data. Along the way, we
demonstrate applications to microphone position calibration, ultrasound
tomography, room reconstruction from echoes and phase retrieval. By spelling
out the essential algorithms, we hope to fast-track the readers in applying
EDMs to their own problems. Matlab code for all the described algorithms, and
to generate the figures in the paper, is available online. Finally, we suggest
directions for further research.

本论文介绍了欧几里得距离矩阵的基本属性以及其在算法设计和应用方面的作用，总结了完成和去噪距离数据方面的相关算法，并给出了多个应用实例，为读者快速应用欧几里得距离矩阵提供帮助。