Bayesian Inference on Matrix Manifolds for Linear Dimensionality Reduction
This provides a novel Bayesian framework for dimensionality reduction, potentially benefiting fields like cognitive neuroscience, though it appears incremental as it extends existing methods to new mathematical structures.
The paper tackles linear dimensionality reduction by reframing it as Bayesian inference on matrix manifolds, using Grassmann and Stiefel manifolds and Hybrid Monte Carlo for posterior sampling, resulting in simpler models and faster speeds for higher-dimensional tasks.
We reframe linear dimensionality reduction as a problem of Bayesian inference on matrix manifolds. This natural paradigm extends the Bayesian framework to dimensionality reduction tasks in higher dimensions with simpler models at greater speeds. Here an orthogonal basis is treated as a single point on a manifold and is associated with a linear subspace on which observations vary maximally. Throughout this paper, we employ the Grassmann and Stiefel manifolds for various dimensionality reduction problems, explore the connection between the two manifolds, and use Hybrid Monte Carlo for posterior sampling on the Grassmannian for the first time. We delineate in which situations either manifold should be considered. Further, matrix manifold models are used to yield scientific insight in the context of cognitive neuroscience, and we conclude that our methods are suitable for basic inference as well as accurate prediction.