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June 2021 Bayesian Inference over the Stiefel Manifold via the Givens Representation
Arya A. Pourzanjani, Richard M. Jiang, Brian Mitchell, Paul J. Atzberger, Linda R. Petzold
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Bayesian Anal. 16(2): 639-666 (June 2021). DOI: 10.1214/20-BA1202

Abstract

We introduce an approach based on the Givens representation for posterior inference in statistical models with orthogonal matrix parameters, such as factor models and probabilistic principal component analysis (PPCA). We show how the Givens representation can be used to develop practical methods for transforming densities over the Stiefel manifold into densities over subsets of Euclidean space. We show how to deal with issues arising from the topology of the Stiefel manifold and how to inexpensively compute the change-of-measure terms. We introduce an auxiliary parameter approach that limits the impact of topological issues. We provide both analysis of our methods and numerical examples demonstrating the effectiveness of the approach. We also discuss how our Givens representation can be used to define general classes of distributions over the space of orthogonal matrices. We then give demonstrations on several examples showing how the Givens approach performs in practice in comparison with other methods.

Acknowledgments

Research reported in this publication was performed by the Systems Biology Coagulopathy of Trauma Program of the US Army Medical Research and Materiel Command under award number W911QY-15-C-0026. The author P.J.A. acknowledges support from research grant DOE ASCR CM4 DE-SC0009254, DOE ASCR PhILMS DE-SC0019246, and NSF DMS - 1616353.

Citation

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Arya A. Pourzanjani. Richard M. Jiang. Brian Mitchell. Paul J. Atzberger. Linda R. Petzold. "Bayesian Inference over the Stiefel Manifold via the Givens Representation." Bayesian Anal. 16 (2) 639 - 666, June 2021. https://doi.org/10.1214/20-BA1202

Information

Published: June 2021
First available in Project Euclid: 3 July 2020

Digital Object Identifier: 10.1214/20-BA1202

Subjects:
Primary: 60K35 , 60K35
Secondary: 60K35

Keywords: dimensionality reduction , orthogonal matrix , Principal Component Analysis , transformation

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Vol.16 • No. 2 • June 2021
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