Silly QR factorization by gradient flow

At last week’s CAM seminar I talked about a “Silly method for QR factorization” (link to slides). Although silly from the point-of-view of efficient numerical algorithms, the method is interesting from a dynamical and geometrical perspective.

The point is that the QR factorization of a real matrix A can be computed as the limit of a gradient flow on the space of multivariate Gaussian distributions with zero mean. The Riemannian structure is the canonical Fisher-Rao metric, and the potential for the gradient is the relative entropy function. What an unexpected connection between QR factorizations and the Riemannian geometry of multivariate Gaussians!

The presentation is based on an excerpt of a recent survey-like paper on the connection between matrix factorizations, optimal transport, and Riemannian geometry.