The approximation of probability measures on compact metric spaces and in particular on Riemannian manifolds by atomic or empirical ones is a classical task in approximation and complexity theory with a wide range of applications. Instead of point measures we are concerned with the approximation by measures supported on Lipschitz curves. Special attention is paid to push-forward measures of Lebesgue measures on the unit interval by such curves. Using the discrepancy as distance between measures, we prove optimal approximation rates in terms of the curve’s length and Lipschitz constant. Having established the theoretical convergence rates, we are interested in the numerical minimization of the discrepancy between a given probability measure and the set of push-forward measures of Lebesgue measures on the unit interval by Lipschitz curves. We present numerical examples for measures on the 2- and 3-dimensional torus, the 2-sphere, the rotation group on \(\mathbb R^3\) and the Grassmannian of all 2-dimensional linear subspaces of \({\mathbb {R}}^4\). Our algorithm of choice is a conjugate gradient method on these manifolds, which incorporates second-order information. For efficient gradient and Hessian evaluations within the algorithm, we approximate the given measures by truncated Fourier series and use fast Fourier transform techniques on these manifolds.

在紧度量空间上，特别是在黎曼流形上，用原子或经验方法对概率度量进行逼近是逼近和复杂性理论中的经典任务，具有广泛的应用。代替点度量，我们关注的是Lipschitz曲线上支持的度量的近似。特别注意通过这种曲线在单位间隔上的Lebesgue测度的前推测度。使用差异作为度量之间的距离，我们证明了曲线的长度和Lipschitz常数的最佳近似率。建立了理论收敛速度后，我们对通过Lipschitz曲线在单位间隔上给定概率测度与Lebesgue测度的前推测度集之间的差异的数值最小化感兴趣。\（\ mathbb R ^ 3 \）和\（{\ mathbb {R}} ^ 4 \）的所有二维线性子空间的Grassmannian 。我们选择的算法是这些流形上的共轭梯度法，其中包含了二阶信息。为了在算法中进行有效的梯度和Hessian评估，我们通过截断傅立叶级数近似给定的度量，并在这些流形上使用快速傅立叶变换技术。