This paper focuses on scattered data fitting problems on spheres. We study
the approximation performance of a class of weighted spectral filter
algorithms, including Tikhonov regularization, Landaweber iteration, spectral
cut-off, and iterated Tikhonov, in fitting noisy data with possibly unbounded
random noise. For the analysis, we develop an integral operator approach that
can be regarded as an extension of the widely used sampling inequality approach
and norming set method in the community of scattered data fitting. After
providing an equivalence between the operator differences and quadrature rules,
we succeed in deriving optimal Sobolev-type error estimates of weighted
spectral filter algorithms. Our derived error estimates do not suffer from the
saturation phenomenon for Tikhonov regularization in the literature,
native-space-barrier for existing error analysis and adapts to different
embedding spaces. We also propose a divide-and-conquer scheme to equip weighted
spectral filter algorithms to reduce their computational burden and present the
optimal approximation error bounds.

该研究聚焦于球体上的散乱数据拟合问题，研究了一类加权谱过滤算法在拟合含有可能无界随机噪声的噪声数据时的近似性能。通过开发一种积分算子方法来分析，我们能够推导出加权谱过滤算法的最优 Sobolev 类型误差估计，同时我们还提出了一种分而治之的方案来降低算法的计算负担，并给出了最优逼近误差界。

球面离散数据拟合的积分算子方法

Integral Operator Approaches for Scattered Data Fitting on Spheres

We study distributed learning with the least squares regularization scheme in
a reproducing kernel Hilbert space (RKHS). By a divide-and-conquer approach,
the algorithm partitions a data set into disjoint data subsets, applies the
least squares regularization scheme to each data subset to produce an output
function, and then takes an average of the individual output functions as a
final global estimator or predictor. We show with error bounds in expectation
in both the $L^2$-metric and RKHS-metric that the global output function of
this distributed learning is a good approximation to the algorithm processing
the whole data in one single machine. Our error bounds are sharp and stated in
a general setting without any eigenfunction assumption. The analysis is
achieved by a novel second order decomposition of operator differences in our
integral operator approach. Even for the classical least squares regularization
scheme in the RKHS associated with a general kernel, we give the best learning
rate in the literature.

使用分布式学习和最小二乘正则化方案，在再生核希尔伯特空间（RKHS）中对分块数据子集应用最小二乘正则化方案生成输出函数，以其平均值作为全局估计器或预测器，具有良好的 $L^2$ 度量和 RKHS 度量误差界限。在我们的积分算子方法中，通过运算符差分的新型二阶分解实现了分析，即使对于与一般核相关联的 RKHS 中的经典最小二乘正则化方案，我们也可以在文献中给出最佳的学习速率。