论文标题
来自噪音散射数据的高维的流形的功能的近似
Approximation of Functions on Manifolds in High Dimension from Noisy Scattered Data
论文作者
论文摘要
在本文中，我们考虑了嵌入高维空间中的低维歧管上函数近似的基本问题，噪声都会影响函数的数据和值。由于维数的诅咒以及噪声的存在，因此在高维情况下，适用于低维度的经典近似方法的有效性较小。我们提出了一种新的近似方法，该方法利用了局部最佳投影（MLOP）方法的优势（由Faigenbaum-Golovin和Levin在2020年引入）以及径向基函数方法（RBF）的优势。该方法是无参数化的，不需要关于歧管的固有维度的知识，可以处理函数值和数据位置中的噪声和异常值，并且直接应用于高维度。我们表明，该方法的复杂性在函数代码域的维度的歧管和平方属性的尺寸中是线性的。随后，我们通过考虑不同的歧管拓扑并将方法的鲁棒性显示到各种噪声水平来证明我们的方法的有效性。
In this paper, we consider the fundamental problem of approximation of functions on a low-dimensional manifold embedded in a high-dimensional space, with noise affecting both in the data and values of the functions. Due to the curse of dimensionality, as well as to the presence of noise, the classical approximation methods applicable in low dimensions are less effective in the high-dimensional case. We propose a new approximation method that leverages the advantages of the Manifold Locally Optimal Projection (MLOP) method (introduced by Faigenbaum-Golovin and Levin in 2020) and the strengths of the method of Radial Basis Functions (RBF). The method is parametrization free, requires no knowledge regarding the manifold's intrinsic dimension, can handle noise and outliers in both the function values and in the location of the data, and is applied directly in the high dimensions. We show that the complexity of the method is linear in the dimension of the manifold and squared-logarithmic in the dimension of the codomain of the function. Subsequently, we demonstrate the effectiveness of our approach by considering different manifold topologies and show the robustness of the method to various noise levels.
