A Retraction Perspective on Low-Rank Dynamics

This talk examines dynamical low-rank approximation (DLRA) [2] through the geometric lens of retractions from Riemannian optimization. The emphasis is on how retractions naturally give rise to time-stepping schemes on fixed-rank manifolds and on how several familiar DLRA integrators can be reinterpreted within that framework [3]. In particular, a KLS retraction — constructed here from the unconventional projector-splitting update [4] — is shown to have a clear geometric relationship to the orthographic retraction [1]. From this perspective we present two complementary integrators: the Accelerated Forward Euler (AFE) method, which exploits an intrinsic acceleration expressed via the Weingarten map, and the Projected Ralston–Hermite (PRH) method, which is based on retraction-driven Hermite interpolation. Under the assumptions discussed, both schemes attain third-order local truncation accuracy, but they display different sensitivities to near-zero singular values and to modeling error. Numerical experiments on benchmark problems, including the differential Lyapunov equation, illustrate these tradeoffs and place the schemes in context with projected Runge–Kutta approaches. Overall, our findings suggest that a retraction-based viewpoint extends DLRA methodology and motivates practical, geometry-aware designs that follow a discretize-in-time-first philosophy.