General Bayesian Inference over the Stiefel Manifold via the Givens Transform

We introduce the Givens Transform, a novel transform between the space of orthonormal matrices and $\mathbb{R}^D$. The Givens Transform allows for the application of any general Bayesian inference algorithm to probabilistic models containing constrained unit-vectors or orthonormal matrix parameters. This includes a variety of matrix factorizations and dimensionality reduction models such as Probabilistic PCA (PPCA), Exponential Family PPCA (BXPCA), and Canonical Correlation Analysis (CCA). While previous Bayesian approaches to these models relied on separate sampling update rules for constrained and unconstrained parameters, the Givens Transform enables the treatment of unit-vectors and orthonormal matrices agnostically as unconstrained parameters. Thus any Bayesian inference algorithm can be used on these models without modification. This opens the door to not just sampling algorithms, but Variational Inference (VI) as well. We illustrate with several examples and supplied code, how the Givens Transform allows end-users to easily build complex models in their favorite Bayesian modeling framework such as Stan, Edward, or PyMC3, a task that was previously intractable due to technical constraints.