Generating learning-friendly representations for points in space is a fundamental and long-standing problem in ML. Recently, multi-scale encoding schemes (such as Space2Vec and NeRF) were proposed to directly encode any point in 2D/3D Euclidean space as a high-dimensional vector, and has been successfully applied to various geospatial prediction and generative tasks. However, all current 2D and 3D location encoders are designed to model point distances in Euclidean space. So when applied to large-scale real-world GPS coordinate datasets, which require distance metric learning on the spherical surface, both types of models can fail due to the map projection distortion problem (2D) and the spherical-to-Euclidean distance approximation error (3D). To solve these problems, we propose a multi-scale location encoder called Sphere2Vec which can preserve spherical distances when encoding point coordinates on a spherical surface. We developed a unified view of distance-reserving encoding on spheres based on the DFS. We also provide theoretical proof that the Sphere2Vec preserves the spherical surface distance between any two points, while existing encoding schemes do not. Experiments on 20 synthetic datasets show that Sphere2Vec can outperform all baseline models on all these datasets with up to 30.8% error rate reduction. We then apply Sphere2Vec to three geo-aware image classification tasks - fine-grained species recognition, Flickr image recognition, and remote sensing image classification. Results on 7 real-world datasets show the superiority of Sphere2Vec over multiple location encoders on all three tasks. Further analysis shows that Sphere2Vec outperforms other location encoder models, especially in the polar regions and data-sparse areas because of its nature for spherical surface distance preservation. Code and data are available at https://gengchenmai.github.io/sphere2vec-website/.

翻译：在空间中为点生成易于学习的表示是机器学习中一个基础且长期存在的问题。近年来，多尺度编码方案（如Space2Vec和NeRF）被提出，可直接将二维/三维欧氏空间中的任意点编码为高维向量，并已成功应用于多种地理空间预测与生成任务。然而，当前所有二维和三维位置编码器均针对欧氏空间中的点距离建模。当应用于需要球形表面距离度量学习的大规模真实GPS坐标数据集时，这两类模型均可能因地图投影畸变问题（二维）及球形到欧氏距离近似误差（三维）而失效。为解决这些问题，我们提出了一种名为Sphere2Vec的多尺度位置编码器，它能在编码球形表面点坐标时保留球面距离。我们基于DFS（深度优先搜索）建立了球面上距离保持编码的统一视角，并理论证明了Sphere2Vec能保持任意两点间的球面距离，而现有编码方案无法做到。在20个合成数据集上的实验表明，Sphere2Vec在所有数据集上均优于所有基线模型，错误率降幅最高达30.8%。随后我们将Sphere2Vec应用于三类地理感知图像分类任务——细粒度物种识别、Flickr图像识别及遥感图像分类。在7个真实数据集上的结果揭示了Sphere2Vec在所有三项任务中相较多种位置编码器的优越性。进一步分析表明，Sphere2Vec因具备球面距离保持特性，尤其在极地及数据稀疏区域中优于其他位置编码器模型。代码与数据可访问https://gengchenmai.github.io/sphere2vec-website/获取。