Many interesting functions arising in applications map into Riemannian manifolds. We present an algorithm, using the manifold exponential and logarithm, for approximating such functions. Our approach extends approximation techniques for functions into linear spaces so that we can upper bound the forward error in terms of a lower bound on the manifold's sectional curvature. Furthermore, when the sectional curvature of a manifold is nonnegative, such as for compact Lie groups, the error is guaranteed to not be worse than in the linear case. We implement the algorithm in a Julia package and apply it to two example problems from Krylov subspaces and dynamic low-rank approximation, respectively. For these examples, the maps are confirmed to be well approximated by our algorithm.

翻译：许多实际应用中涌现的函数都映射到黎曼流形中。本文提出一种基于流形指数映射与对数映射的算法，用于逼近此类函数。该方法将线性空间中的函数逼近技术进行推广，从而能够根据流形截面曲率的下界给出前向误差的上界。更关键的是，当流形截面曲率为非负时（例如紧致李群），其误差保证不会劣于线性情形。我们在Julia语言包中实现了该算法，并将其分别应用于Krylov子空间与动态低秩逼近两个典型问题。实验结果表明，我们的算法能够有效逼近这些实例中的映射函数。