Bayesian Manifold Regression

TL;DR

This paper develops a Bayesian nonparametric regression method for high-dimensional data concentrated on a low-dimensional manifold, achieving optimal rates without explicitly estimating the manifold.

Contribution

It introduces a Gaussian process regression approach that is computationally feasible and theoretically optimal for manifold-based high-dimensional regression problems.

Findings

Achieves minimax optimal adaptive rates in regression estimation.

Does not require explicit manifold estimation, simplifying computations.

Demonstrates effectiveness through finite sample data analysis.

Abstract

There is increasing interest in the problem of nonparametric regression with high-dimensional predictors. When the number of predictors $D$ is large, one encounters a daunting problem in attempting to estimate a $D$-dimensional surface based on limited data. Fortunately, in many applications, the support of the data is concentrated on a $d$-dimensional subspace with $d \ll D$. Manifold learning attempts to estimate this subspace. Our focus is on developing computationally tractable and theoretically supported Bayesian nonparametric regression methods in this context. When the subspace corresponds to a locally-Euclidean compact Riemannian manifold, we show that a Gaussian process regression approach can be applied that leads to the minimax optimal adaptive rate in estimating the regression function under some conditions. The proposed model bypasses the need to estimate the manifold, and can be implemented using standard algorithms for posterior computation in Gaussian processes. Finite sample performance is illustrated in an example data analysis.