Work

Low-Rank Plus Diagonal Approximations for Riccati-Like Matrix Differential Equations

Year: 2024

Type: article

Abstract: . We consider the problem of computing tractable approximations of time-dependent [Formula: see text] large positive semidefinite (PSD) matrices defined as solutions of a matrix differential equation. We propose to use “low-rank plus diagonal” PSD matrices as approximations that can be stored with a memory cost being linear in the high dimension [Formula: see text]. To constrain the solution of the differential equation to remain in that subset, we project the derivative at all times onto the tangent space to the subset, following the methodology of dynamical low-rank approximation. We derive a closed-form formula for the projection and show that after some manipulations, it can be computed with a numerical cost being linear in [Formula: see text], allowing for tractable implementation. Contrary to previous approaches based on pure low-rank approximations, the addition of the diagonal term allows for our approximations to be invertible matrices that can moreover be inverted with linear cost in [Formula: see text]. We apply the technique to Riccati-like equations, then to two particular problems: first, a low-rank approximation to our recent Wasserstein gradient flow for Gaussian approximation of posterior distributions in approximate Bayesian inference and, second, a novel low-rank approximation of the Kalman filter for high-dimensional systems. Numerical simulations illustrate the results.

Source: SIAM Journal on Matrix Analysis and Applications

DOI: 10.1137/23m1587610

Authors: Silvère Bonnabel, Marc Lambert, Francis Bach