Quantiles on global non-positive curvature spaces

Abstract We propose a notion of geometric quantiles on Hadamard spaces, or 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, we investigate asymptotic properties, 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. We detail several potential uses for these quantiles—defining measures of distributional characteristics, detecting outliers via a transformation–retransformation procedure, and testing the equality of distributions—and demonstrate their application through simulations and real biological data.