Comparing spherical probability distributions is of great interest in various
fields, including geology, medical domains, computer vision, and deep
representation learning. The utility of optimal transport-based distances, such
as the Wasserstein distance, for comparing probability measures has spurred
active research in developing computationally efficient variations of these
distances for spherical probability measures. This paper introduces a
high-speed and highly parallelizable distance for comparing spherical measures
using the stereographic projection and the generalized Radon transform, which
we refer to as the Stereographic Spherical Sliced Wasserstein (S3W) distance.
We carefully address the distance distortion caused by the stereographic
projection and provide an extensive theoretical analysis of our proposed metric
and its rotationally invariant variation. Finally, we evaluate the performance
of the proposed metrics and compare them with recent baselines in terms of both
speed and accuracy through a wide range of numerical studies, including
gradient flows and self-supervised learning.

本论文介绍了一种用于比较球形概率测度的高速、高度并行化的距离度量方法 ——Stereographic Spherical Sliced Wasserstein (S3W) 距离，并通过广泛的理论分析和数值研究评估了该距离与最近基准方法在速度和准确性方面的表现。

立体地球切片瓦塞尔斯坦距离

Stereographic Spherical Sliced Wasserstein Distances

Graph Neural Networks leverage the connectivity structure of graphs as an
inductive bias. Latent graph inference focuses on learning an adequate graph
structure to diffuse information on and improve the downstream performance of
the model. In this work we employ stereographic projections of the hyperbolic
and spherical model spaces, as well as products of Riemannian manifolds, for
the purpose of latent graph inference. Stereographically projected model spaces
achieve comparable performance to their non-projected counterparts, while
providing theoretical guarantees that avoid divergence of the spaces when the
curvature tends to zero. We perform experiments on both homophilic and
heterophilic graphs.

本文采用超球面和双曲线模型空间的立体投影以及黎曼流形积的方法进行潜在图推理，实现与非投影对应物相当的性能，同时提供理论保证，避免当曲率趋近于零时空间发散的问题，并在同构和异构图上进行了实验。

潜在图推断的模型空间投影

Projections of Model Spaces for Latent Graph Inference

Wide-angle portraits often enjoy expanded views. However, they contain
perspective distortions, especially noticeable when capturing group portrait
photos, where the background is skewed and faces are stretched. This paper
introduces the first deep learning based approach to remove such artifacts from
freely-shot photos. Specifically, given a wide-angle portrait as input, we
build a cascaded network consisting of a LineNet, a ShapeNet, and a transition
module (TM), which corrects perspective distortions on the background, adapts
to the stereographic projection on facial regions, and achieves smooth
transitions between these two projections, accordingly. To train our network,
we build the first perspective portrait dataset with a large diversity in
identities, scenes and camera modules. For the quantitative evaluation, we
introduce two novel metrics, line consistency and face congruence. Compared to
the previous state-of-the-art approach, our method does not require camera
distortion parameters. We demonstrate that our approach significantly
outperforms the previous state-of-the-art approach both qualitatively and
quantitatively.

本研究介绍了第一个基于深度学习的方法，用于去除自由拍摄的广角人像照片中的透视失真和形变等因素，建立了一个包括 LineNet，ShapeNet 和 transition module（TM）的级联网络，并提出了两个新的度量标准：线条一致性和面部一致性，相比之前的最先进方法，本方法不需要相机失真参数，能够在身份、场景和相机模块的大多数多样性方面得到较好的性能。