Monte Carlo Simulation on the Stiefel Manifold via Polar Expansion

Motivated by applications to Bayesian inference for statistical models with orthogonal matrix parameters, we present (Formula presented.) a general approach to Monte Carlo simulation from probability distributions on the Stiefel manifold. To bypass many of the well-established challenges of simulating from the distribution of a random orthogonal matrix (Formula presented.) we construct a distribution for an unconstrained random matrix X such that (Formula presented.) the orthogonal component of the polar decomposition of (Formula presented.) is equal in distribution to (Formula presented.) The distribution of X is amenable to Markov chain Monte Carlo (MCMC) simulation using standard methods, and an approximation to the distribution of Q can be recovered from a Markov chain on the unconstrained space. When combined with modern MCMC software, polar expansion allows for routine and flexible posterior inference in models with orthogonal matrix parameters. We find that polar expansion with adaptive Hamiltonian Monte Carlo is an order of magnitude more efficient than competing MCMC approaches in a benchmark protein interaction network application. We also propose a new approach to Bayesian functional principal component analysis which we illustrate in a meteorological time series application. Supplementary materials for this article are available online.

Duke Scholars

Author David B. Dunson Statistical Science