论文标题
双曲线切成薄片 - 通过测量和霍斯氏预测
Hyperbolic Sliced-Wasserstein via Geodesic and Horospherical Projections
论文作者
论文摘要
它已显示出对许多类型的数据有益的，这些数据呈现出嵌入双曲线空间中的基本层次结构。因此，许多机器学习工具扩展到了此类空间，但是只有很少的差异可以比较存在这些空间上定义的概率分布。在可能的候选人中，最佳的运输距离在此类riemannian流形上得到了很好的定义，并具有强大的理论特性，但遭受了高计算成本。在欧几里得空间上，切成薄片的距离距离在一个维度上利用瓦斯汀距离的封闭形式，在一个维度上具有更高的计算效率，但在双曲线空间上不容易获得。在这项工作中，我们提议得出新型的双曲线切片式差异。这些构造使用沿霍斯莱斯或大地测量学的潜在测量学上的预测。我们在相关双曲表示的不同任务上研究和比较它们，例如采样或图像分类。
It has been shown beneficial for many types of data which present an underlying hierarchical structure to be embedded in hyperbolic spaces. Consequently, many tools of machine learning were extended to such spaces, but only few discrepancies to compare probability distributions defined over those spaces exist. Among the possible candidates, optimal transport distances are well defined on such Riemannian manifolds and enjoy strong theoretical properties, but suffer from high computational cost. On Euclidean spaces, sliced-Wasserstein distances, which leverage a closed-form of the Wasserstein distance in one dimension, are more computationally efficient, but are not readily available on hyperbolic spaces. In this work, we propose to derive novel hyperbolic sliced-Wasserstein discrepancies. These constructions use projections on the underlying geodesics either along horospheres or geodesics. We study and compare them on different tasks where hyperbolic representations are relevant, such as sampling or image classification.
