The Annals of Statistics
- Ann. Statist.
- Volume 44, Number 2 (2016), 876-905.
Bayesian manifold regression
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 a data analysis example.
Ann. Statist., Volume 44, Number 2 (2016), 876-905.
Received: December 2014
Revised: September 2015
First available in Project Euclid: 17 March 2016
Permanent link to this document
Digital Object Identifier
Mathematical Reviews number (MathSciNet)
Zentralblatt MATH identifier
Yang, Yun; Dunson, David B. Bayesian manifold regression. Ann. Statist. 44 (2016), no. 2, 876--905. doi:10.1214/15-AOS1390. https://projecteuclid.org/euclid.aos/1458245738
- Reviews of geometric properties and proofs of Theorems 2.1, 2.2, 2.4 and 3.2. Concepts and results in differential and Riemannian geometry were reviewed in Section 7, where new results are included with proofs. Then proofs of Theorems 2.1, 2.2, 2.4 and 3.2 are provided in Section 8.