Many of the primal ingredients of convex optimization extend naturally from Euclidean to Hadamard spaces $\unicode{x2014}$ nonpositively curved metric spaces like Euclidean, Hilbert, and hyperbolic spaces, metric trees, and more general CAT(0) cubical complexes. Linear structure, however, and the duality theory it supports are absent. Nonetheless, we introduce a new type of subgradient for convex functions on Hadamard spaces, based on Busemann functions. This notion supports a splitting subgradient method with guaranteed complexity bounds. In particular, the algorithm solves $p$-mean problems in general Hadamard spaces: we illustrate by computing medians in BHV tree space.

翻译：凸优化的许多基本要素可以自然地由欧几里得空间推广至Hadamard空间——即非正曲率度量空间，如欧几里得空间、希尔伯特空间、双曲空间、度量树，以及更一般的CAT(0)立方复形。然而，线性结构及其所支撑的对偶理论在此类空间中并不存在。尽管如此，我们基于Busemann函数为Hadamard空间上的凸函数引入了一种新型次梯度概念。这一概念支持具有可证明复杂度界的分裂次梯度方法。特别地，该算法可求解一般Hadamard空间中的$p$-均值问题：我们通过计算BHV树空间中的中位数来展示其应用。