Bayesian Compressed Regression

Abstract

As an alternative to variable selection or shrinkage in high-dimensional regression, we propose to randomly compress the predictors prior to analysis. This dramatically reduces storage and computational bottlenecks, performing well when the predictors can be projected to a low-dimensional linear subspace with minimal loss of information about the response. As opposed to existing Bayesian dimensionality reduction approaches, the exact posterior distribution conditional on the compressed data is available analytically, speeding up computation by many orders of magnitude while also bypassing robustness issues due to convergence and mixing problems with MCMC. Model averaging is used to reduce sensitivity to the random projection matrix, while accommodating uncertainty in the subspace dimension. Strong theoretical support is provided for the approach by showing near parametric convergence rates for the predictive density in the large p small n asymptotic paradigm. Practical performance relative to competitors is illustrated in simulations and real data applications.

• Compressed sensing
• Data compression
• Dimensionality reduction
• Large p, small n
• Random projection
• Sparsity
• Sufficient dimension reduction.

Additional information

Notes on contributors

Rajarshi Guhaniyogi

Rajarshi Guhaniyogi is Assistant Professor, Department of Applied Mathematics & Statistics, Baskin School of Engineering, SOE2, University of California Santa Cruz, 1156 High St., Santa Cruz, CA 95064 (E-mail: [email protected]). David B. Dunson is Professor, Department of Statistical Science, 219A Old Chemistry Building, Box 90251, Duke University, Durham, NC 27708-0251 (E-mail: [email protected]).

David B. Dunson

Rajarshi Guhaniyogi is Assistant Professor, Department of Applied Mathematics & Statistics, Baskin School of Engineering, SOE2, University of California Santa Cruz, 1156 High St., Santa Cruz, CA 95064 (E-mail: [email protected]). David B. Dunson is Professor, Department of Statistical Science, 219A Old Chemistry Building, Box 90251, Duke University, Durham, NC 27708-0251 (E-mail: [email protected]).