论文标题
基于随机矩阵的方案，用于稳定且稳健的非参数和功能回归估计器
Random matrices based schemes for stable and robust nonparametric and functional regression estimators
论文作者
论文摘要
在这项工作的第一部分中，我们开发了一种解决非参数回归问题的新计划。这是从某些随机点给出的近似值的知识，可能是较低的常规函数和差异的近似值。我们提出的方案是基于随机投影矩阵的伪内的使用，并结合了Jacobi多项式系统的某些特定属性，以及某些正定随机矩阵的某些属性。就执行时间而言，该方案具有稳定，健壮，准确且相当快的优势。特别是，我们提供了$ L_2 $以及我们建议的非参数回归估算器的$ L_2- $风险错误。此外，与大多数现有的非参数回归估计器不同，我们提出的估计器不需要额外的正规化步骤。虽然，该估计器最初设计用于与uni-bariate I.I.D.的随机采样集一起使用。在Beta分布之后，随机变量，我们表明它仍然适用于广泛的采样分布定律。此外，我们简要描述了如何对估计器进行调整，以处理随机采样集的多元案例。在这项工作的第二部分中，我们扩展了随机的伪内方案技术，以构建一个稳定而准确的估计器来解决线性功能回归（LFR）问题。一种二元分解方法用于构建有关LFR问题的最后一个稳定估计器。 Alaso，我们提出的LFR估计器给出了$ L_2- $风险错误。最后，通过各种数值模拟说明了两个提出的估计器的性能。特别是，使用真实的数据集来说明我们的非参数回归估算器的性能。
In the first part of this work, we develop a novel scheme for solving nonparametric regression problems. That is the approximation of possibly low regular and noised functions from the knowledge of their approximate values given at some random points. Our proposed scheme is based on the use of the pseudo-inverse of a random projection matrix, combined with some specific properties of the Jacobi polynomials system, as well as some properties of positive definite random matrices. This scheme has the advantages to be stable, robust, accurate and fairly fast in terms of execution time. In particular, we provide an $L_2$ as well as an $L_2-$risk errors of our proposed nonparametric regression estimator. Moreover and unlike most of the existing nonparametric regression estimators, no extra regularization step is required by our proposed estimator. Although, this estimator is initially designed to work with random sampling set of uni-variate i.i.d. random variables following a Beta distribution, we show that it is still works for a wide range of sampling distribution laws. Moreover, we briefly describe how our estimator can be adapted in order to handle the multivariate case of random sampling sets. In the second part of this work, we extend the random pseudo-inverse scheme technique to build a stable and accurate estimator for solving linear functional regression (LFR) problems. A dyadic decomposition approach is used to construct this last stable estimator for the LFR problem. Alaso, we give an $L_2-$risk error of our proposed LFR estimator. Finally, the performance of the two proposed estimators are illustrated by various numerical simulations. In particular, a real dataset is used to illustrate the performance of our nonparametric regression estimator.
