论文标题
通过筛子的方法在张量产品空间中回归
Regression in Tensor Product Spaces by the Method of Sieves
论文作者
论文摘要
估计条件平均值（将一组特征与感兴趣的结果联系起来）是一项基本统计任务。尽管对灵活的非参数程序有吸引力，但在许多经典的非参数函数空间（例如，多元SOBOLEV空间）中的有效估计可能非常困难 - 无论是统计和计算上 - 尤其是在特征数量很大时。在本文中，我们介绍了（受惩罚）非参数张量产品空间回归的筛子估计器：这些空间更适合多变量回归，并让我们避免避免尺寸的诅咒。我们的估计器可以轻松地应用于多元非参数问题，并具有吸引人的统计和计算特性。此外，它们可以有效利用其他结构（例如特征稀疏性）。在本手稿中，我们提供理论保证，表明我们的估计器的预测性能在维度方面有利。此外，我们还提出了数值示例，以将所提出的估计器的有限样本性能与几种流行的机器学习方法进行比较。
Estimation of a conditional mean (linking a set of features to an outcome of interest) is a fundamental statistical task. While there is an appeal to flexible nonparametric procedures, effective estimation in many classical nonparametric function spaces (e.g., multivariate Sobolev spaces) can be prohibitively difficult -- both statistically and computationally -- especially when the number of features is large. In this paper, we present (penalized) sieve estimators for regression in nonparametric tensor product spaces: These spaces are more amenable to multivariate regression, and allow us to, in-part, avoid the curse of dimensionality. Our estimators can be easily applied to multivariate nonparametric problems and have appealing statistical and computational properties. Moreover, they can effectively leverage additional structures such as feature sparsity. In this manuscript, we give theoretical guarantees, indicating that the predictive performance of our estimators scale favorably in dimension. In addition, we also present numerical examples to compare the finite-sample performance of the proposed estimators with several popular machine learning methods.
