2013/05/03 by Yun Yang, David B. Dunson, Yang, Yun +1 · 2 citations
Computer Science · Mathematics · #Bayesian Methods and Mixture Models #FOS: Mathematics #Gaussian Processes and Bayesian Inference #Statistical Methods and Inference #Statistics Theory (math.ST) #math.ST #stat.TH
paper · pdf · doi:10.48550/arxiv.1305.0617
We added a new section (Section 3) with two empirical Bayes approaches to make our method adaptive to the intrinsic dimension of the manifold with theoretical guarantees. We also rearranged the paper and deleted a subsection in the previous version that lacks rigorous theoretical support
openalex publication_date 2013/05/03 · arxiv created 2014/06/16 · arxiv updated 2014/06/17 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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 ≪ 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.