# Monte Carlo simulation on the Stiefel manifold via polar expansion

## TL;DR

This paper introduces polar expansion, a novel Monte Carlo simulation method for distributions on the Stiefel manifold, enabling efficient Bayesian inference for models with orthogonal matrix parameters.

## Contribution

The paper presents polar expansion, a new approach that simplifies sampling from distributions on the Stiefel manifold using MCMC on an unconstrained space, improving efficiency.

## Key findings

- Polar expansion with HMC is an order of magnitude more efficient than existing methods.
- The method enables routine Bayesian inference for models with orthogonal matrix parameters.
- Application to protein networks and meteorological data demonstrates practical utility.

## Abstract

Motivated by applications to Bayesian inference for statistical models with orthogonal matrix parameters, we present $\textit{polar expansion},$ 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 $\boldsymbol{Q},$ we construct a distribution for an unconstrained random matrix $\boldsymbol{X}$ such that $\boldsymbol{Q}_X,$ the orthogonal component of the polar decomposition of $\boldsymbol{X},$ is equal in distribution to $\boldsymbol{Q}.$ The distribution of $\boldsymbol{X}$ is amenable to Markov chain Monte Carlo (MCMC) simulation using standard methods, and an approximation to the distribution of $\boldsymbol{Q}$ 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 components analysis which we illustrate in a meteorological time series application.