A Spectral Series Approach to High-Dimensional Nonparametric Regression
This work addresses the challenge of generalizing sparse techniques to complex, high-dimensional data like images and trajectories, though it appears incremental as it ties together existing ideas from kernel learning and Fourier methods.
The authors tackled the problem of making fast and reliable inferences for high-dimensional data with nonlinear structure by developing an orthogonal series estimator that adapts to data geometry, resulting in systematic performance comparisons with state-of-the-art methods on simulated and real-world data.
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.