We consider the problem of approximating a function f from an Euclidean domain to a manifold M by scattered samples $$(f(\xi _i))_{i\in \mathcal {I}}$$(f(ξi))i∈I, where the data sites $$(\xi _i)_{i\in \mathcal {I}}$$(ξi)i∈I are assumed to be locally close but can otherwise be far apart points scattered throughout the domain. We introduce a natural approximant based on combining the moving least square method and the Karcher mean. We prove that the proposed approximant inherits the accuracy order and the smoothness from its linear counterpart. The analysis also tells us that the use of Karcher’s mean (dependent on a Riemannian metric and the associated exponential map) is inessential and one can replace it by a more general notion of ‘center of mass’ based on a general retraction on the manifold. Consequently, we can substitute the Karcher mean by a more computationally efficient mean. We illustrate our work with numerical results which confirm our theoretical findings.

中文翻译：

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分散流形值数据近似

我们考虑通过散布样本将函数 f 从欧几里得域逼近到流形 M $$(f(\xi _i))_{i\in \mathcal {I}}$$(f(ξi))i ∈I，其中数据站点 $$(\xi _i)_{i\in \mathcal {I}}$$(ξi)i∈I 被假定为局部接近，但也可以远离分散在整个域中的点. 我们引入了一种基于结合移动最小二乘法和 Karcher 均值的自然逼近。我们证明所提出的近似继承了其线性对应项的准确度顺序和平滑度。分析还告诉我们，使用 Karcher 均值（取决于黎曼度量和相关的指数映射）是无关紧要的，可以用基于流形上的一般回缩的更一般的“质心”概念来代替它。最后，我们可以用计算效率更高的均值代替 Karcher 均值。我们用数值结果来说明我们的工作，这证实了我们的理论发现。