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.
翻译:在本文中,我们考虑了在高维空间嵌入的低维元体上近似功能的根本问题,其中的噪音影响到该功能的数据和值。由于维度的诅咒以及噪音的存在,适用于低维的古典近似方法在高维情况下不太有效。我们提出了一种新的近似方法,该方法利用了Manixy地方最佳投影(MLOP)方法(Faigenbaum-Golovin和Levin于2020年推出)的优势,以及Radial Basic函数(RBF)方法的优势。该方法是免费的,不需要了解元的内在维度,无法在功能值和数据位置上处理噪音和外在数据位置上直接应用于高维度。我们表明,该方法的复杂性是线性,在功能的共和正方对数方面是线性的。随后,我们通过考虑不同的多元表层和显示方法的稳健度,展示了我们的方法的有效性。