This paper develops a notion of geometric quantiles on Hadamard spaces, also known as global non-positive curvature spaces. After providing some definitions and basic properties, including scaled isometry equivariance and a necessary condition on the gradient of the quantile loss function at quantiles on Hadamard manifolds, we investigate asymptotic properties of sample quantiles on Hadamard manifolds, such as strong consistency and joint asymptotic normality. We provide a detailed description of how to compute quantiles using a gradient descent algorithm in hyperbolic space and, in particular, an explicit formula for the gradient of the quantile loss function, along with experiments using simulated and real single-cell RNA sequencing data.

翻译：本文提出了哈达玛德空间（即全局非正曲率空间）上的几何分位数概念。在给出相关定义及基本性质（包括缩放等距同变性以及哈达玛德流形上分位数损失函数梯度存在的必要条件）后，我们研究了哈达玛德流形上样本分位数的渐近性质，如强相合性和联合渐近正态性。详细描述了如何在双曲空间中利用梯度下降算法计算分位数，特别给出了分位数损失函数梯度的显式公式，并基于模拟数据和真实单细胞RNA测序数据开展了实验验证。