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：本文在Hadamard空间（即全局非正曲率空间）上建立了几何分位数的概念。在给出若干定义与基本性质（包括尺度等距协变性及Hadamard流形上分位损失函数在分位数处的梯度必要条件）之后，我们研究了Hadamard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。我们详细描述了如何在双曲空间中使用梯度下降算法计算分位数，并特别给出了分位损失函数梯度的显式公式，同时提供了基于模拟数据和真实单细胞RNA测序数据的实验验证。