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Bayesian manifold regression

Publication ,  Journal Article
Yang, Y; Dunson, DB
Published in: Annals of Statistics
April 1, 2016

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 aD-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 a data analysis example.

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Published In

Annals of Statistics

DOI

ISSN

0090-5364

Publication Date

April 1, 2016

Volume

44

Issue

2

Start / End Page

876 / 905

Related Subject Headings

  • Statistics & Probability
  • 4905 Statistics
  • 3802 Econometrics
  • 1403 Econometrics
  • 0104 Statistics
  • 0102 Applied Mathematics
 

Citation

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Yang, Y., & Dunson, D. B. (2016). Bayesian manifold regression. Annals of Statistics, 44(2), 876–905. https://doi.org/10.1214/15-AOS1390
Yang, Y., and D. B. Dunson. “Bayesian manifold regression.” Annals of Statistics 44, no. 2 (April 1, 2016): 876–905. https://doi.org/10.1214/15-AOS1390.
Yang Y, Dunson DB. Bayesian manifold regression. Annals of Statistics. 2016 Apr 1;44(2):876–905.
Yang, Y., and D. B. Dunson. “Bayesian manifold regression.” Annals of Statistics, vol. 44, no. 2, Apr. 2016, pp. 876–905. Scopus, doi:10.1214/15-AOS1390.
Yang Y, Dunson DB. Bayesian manifold regression. Annals of Statistics. 2016 Apr 1;44(2):876–905.

Published In

Annals of Statistics

DOI

ISSN

0090-5364

Publication Date

April 1, 2016

Volume

44

Issue

2

Start / End Page

876 / 905

Related Subject Headings

  • Statistics & Probability
  • 4905 Statistics
  • 3802 Econometrics
  • 1403 Econometrics
  • 0104 Statistics
  • 0102 Applied Mathematics