Electronic Journal of Statistics

A spectral series approach to high-dimensional nonparametric regression

Ann B. Lee and Rafael Izbicki

Full-text: Open access

Abstract

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 manifold learning, 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.

Article information

Source
Electron. J. Statist., Volume 10, Number 1 (2016), 423-463.

Dates
Received: June 2015
First available in Project Euclid: 24 February 2016

Permanent link to this document
https://projecteuclid.org/euclid.ejs/1456322681

Digital Object Identifier
doi:10.1214/16-EJS1112

Mathematical Reviews number (MathSciNet)
MR3466189

Zentralblatt MATH identifier
1332.62133

Subjects
Primary: 62G08: Nonparametric regression

Keywords
High-dimensional inference orthogonal series regression data-driven basis Mercer kernel manifold learning eigenmaps

Citation

Lee, Ann B.; Izbicki, Rafael. A spectral series approach to high-dimensional nonparametric regression. Electron. J. Statist. 10 (2016), no. 1, 423--463. doi:10.1214/16-EJS1112. https://projecteuclid.org/euclid.ejs/1456322681


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