We present an algorithm for approximating a function defined over a $d$-dimensional manifold utilizing only noisy function values at locations sampled from the manifold with noise. To produce the approximation we do not require knowledge about the local geometry of the manifold or its local parameterizations. We do require, however, knowledge regarding the manifold’s intrinsic dimension $d$. We use the Manifold Moving Least-Squares approach of Sober and Levin (2019) to reconstruct the atlas of charts and the approximation is built on top of those charts. The resulting approximant is shown to be a function defined over a neighborhood of a manifold, approximating the originally sampled manifold. In other words, given a new point, located near the manifold, the approximation can be evaluated directly on that point. We prove that our construction yields a smooth function, and in case of noiseless samples the approximation order is $\mathcal{O}\left(h^{m+1}\right)$, where $h$ is a local density of sample parameter (i.e., the fill distance) and $m$ is the degree of a local polynomial approximation, used in our algorithm. In addition, the proposed algorithm has linear time complexity with respect to the ambient space’s dimension. Thus, we are able to avoid the computational complexity, commonly encountered in high dimensional approximations, without having to perform non-linear dimension reduction, which inevitably introduces distortions to the geometry of the data. Additionally, we show numerically that our approach compares favorably to some well-known approaches for regression over manifolds.

我们提出一种算法，用于近似定义一个 $d$维歧管仅在从歧管采样的带噪声的位置使用噪声函数值。为了产生近似值，我们不需要关于流形的局部几何或其局部参数化的知识。但是，我们确实需要有关流形固有尺寸的知识$d$。我们使用Sober和Levin（2019）的歧管移动最小二乘法（Manifold Moving Least-Squares）方法重构图表集，并在这些图表之上建立近似值。所得的近似值显示为在歧管附近定义的函数，近似原始采样的歧管。换句话说，给定一个新点，该点位于流形附近，可以直接在该点上评估近似值。我们证明了我们的构造产生了平滑函数，并且在无噪声样本的情况下，近似阶为$\mathcal{Ø}（H^{米+\mathrm{1个}}）$，在哪里 $H$ 是样品参数的局部密度（即填充距离），并且 $米$是我们算法中使用的局部多项式逼近度。另外，所提出的算法相对于周围空间的尺寸具有线性时间复杂度。因此，我们能够避免在高维近似中经常遇到的计算复杂性，而不必执行非线性维降，这不可避免地将失真引入数据的几何形状。此外，我们通过数字显示，与通过流形进行回归的一些知名方法相比，我们的方法具有优势。