Function approximation based on data drawn randomly from an unknown
distribution is an important problem in machine learning. In contrast to the
prevalent paradigm of solving this problem by minimizing a loss functional, we
have given a direct one-shot construction together with optimal error bounds
under the manifold assumption; i.e., one assumes that the data is sampled from
an unknown sub-manifold of a high dimensional Euclidean space. A great deal of
research deals with obtaining information about this manifold, such as the
eigendecomposition of the Laplace-Beltrami operator or coordinate charts, and
using this information for function approximation. This two step approach
implies some extra errors in the approximation stemming from basic quantities
of the data in addition to the errors inherent in function approximation. In
Neural Networks, 132:253268, 2020, we have proposed a one-shot direct method to
achieve function approximation without requiring the extraction of any
information about the manifold other than its dimension. However, one cannot
pin down the class of approximants used in that paper.
In this paper, we view the unknown manifold as a sub-manifold of an ambient
hypersphere and study the question of constructing a one-shot approximation
using the spherical polynomials based on the hypersphere. Our approach does not
require preprocessing of the data to obtain information about the manifold
other than its dimension. We give optimal rates of approximation for relatively
"rough" functions.

基于 Hypersphere 上的球面多项式，无需预处理数据即可构建一次性逼近，并给出了相对 “粗糙” 函数的最佳逼近速率。

无需流行学习的流形上的学习

Learning on manifolds without manifold learning

The purpose of this paper is to construct universal, auto--adaptive,
localized, linear, polynomial (-valued) operators based on scattered data on
the (hyper--)sphere $\SS^q$ ($q\ge 2$). The approximation and localization
properties of our operators are studied theoretically in deterministic as well
as probabilistic settings. Numerical experiments are presented to demonstrate
their superiority over traditional least squares and discrete Fourier
projection polynomial approximations. An essential ingredient in our
construction is the construction of quadrature formulas based on scattered
data, exact for integrating spherical polynomials of (moderately) high degree.
Our formulas are based on scattered sites; i.e., in contrast to such well known
formulas as Driscoll--Healy formulas, we need not choose the location of the
sites in any particular manner. While the previous attempts to construct such
formulas have yielded formulas exact for spherical polynomials of degree at
most 18, we are able to construct formulas exact for spherical polynomials of
degree 178.

本文旨在基于（超）球面 S^q（q≥2）上的散乱数据构建通用的、自适应的、本地化的、线性的、多项式（值）算子。我们研究了我们算子的逼近和定位性质，从确定性和概率的角度进行了研究。数值实验表明，我们的算子相对于传统的最小二乘和离散傅里叶投影多项式逼近方法优越。我们构造四面体多项式的积分公式的一个基本要素是基于散乱数据，确保其准确性，可以有效地积分（适度）高次数的球面函数。而我们的公式是基于散乱站点的；即，与诸如 Driscoll-Healy 公式等众所周知的公式不同，我们不需要以任何特定的方式选择站点的位置。虽然之前试图构造这种公式的尝试只能得到对不超过 18 次的球面多项式精确的公式，但我们能够构造出对球面多项式精确的公式，其次数为 178。