标题： 高维非参数回归的谱系列方法 显示英文标题

标题： A Spectral Series Approach to High-Dimensional Nonparametric Regression

摘要： 现代统计学中的一个重要问题是，如何对复杂、高维的数据进行快速且可靠的推断。尽管稀疏技术受到了广泛关注，但目前的方法在处理具有非线性结构的数据时泛化能力较差。在这项工作中，我们提出了一种正交级数估计器，用于复杂的复合预测变量（如自然图像、星系光谱、轨迹和电影）。 我们的级数方法结合了核机器学习和傅里叶方法的思想。我们将未知回归函数在数据上用基于核算子的特征函数展开，并利用基底相对于潜在数据分布P的正交性来加速计算和参数调整。如果核选择得当，那么特征函数可以自适应于数据的内在几何结构和维度。对于具有变化带宽的径向核，我们提供了理论保证，并将回归函数相对于P的光滑性与特征基的稀疏性联系起来。 最后，使用模拟数据和真实数据，我们系统地比较了频谱级数方法与经典核平滑、k-最近邻回归、核岭回归以及最先进的流形和局部回归方法的性能。 显示英文摘要

摘要： A key question in modern statistics is how to make fast and reliable inferences for complex, high-dimensional data. While there has been much interest in sparse techniques, current methods do not generalize well to data with nonlinear structure. In this work, we present an orthogonal series estimator for predictors that are complex aggregate objects, such as natural images, galaxy spectra, trajectories, and movies. Our series approach ties together ideas from kernel machine learning, and Fourier methods. We expand the unknown regression on the data in terms of the eigenfunctions of a kernel-based operator, and we take advantage of orthogonality of the basis with respect to the underlying data distribution, P, to speed up computations and tuning of parameters. If the kernel is appropriately chosen, then the eigenfunctions adapt to the intrinsic geometry and dimension of the data. We provide theoretical guarantees for a radial kernel with varying bandwidth, and we relate smoothness of the regression function with respect to P to sparsity in the eigenbasis. Finally, using simulated and real-world data, we systematically compare the performance of the spectral series approach with classical kernel smoothing, k-nearest neighbors regression, kernel ridge regression, and state-of-the-art manifold and local regression methods.