Quantile regression (QR) is a statistical tool for distribution-free estimation of conditional quantiles of a target variable given explanatory features. QR is limited by the assumption that the target distribution is univariate and defined on an Euclidean domain. Although the notion of quantiles was recently extended to multi-variate distributions, QR for multi-variate distributions on manifolds remains underexplored, even though many important applications inherently involve data distributed on, e.g., spheres (climate measurements), tori (dihedral angles in proteins), or Lie groups (attitude in navigation). By leveraging optimal transport theory and the notion of $c$-concave functions, we meaningfully define conditional vector quantile functions of high-dimensional variables on manifolds (M-CVQFs). Our approach allows for quantile estimation, regression, and computation of conditional confidence sets. We demonstrate the approach's efficacy and provide insights regarding the meaning of non-Euclidean quantiles through preliminary synthetic data experiments.

运用最优输运理论和$c$-凹函数的思想，我们有意义地定义了高维空间流形上的条件向量分位函数（M-CVQFs），从而实现了流形上多元分布的分布自由估计、回归及条件置信集计算。通过初步的合成数据实验，展现该方法的功效和关于非欧几里德分位数的含义的一些见解。

流形上向量分位数回归