Geodesic strong convexity does not imply forward invariance under gradient flow on SO(3): a certified counterexample

What the math gives to ML

The paper provides a certified counterexample showing that geodesic strong convexity and an interior minimizer do not imply forward invariance of a geodesic trust region under gradient flow on \(\mathrm{SO}(3)\). Its transferable mechanism is a separation between curvature relative to the minimizer and the boundary-normal component required by the viability condition: a strongly convex objective can still point outward at the boundary. Neural-network training on Lie-group parameters, latent rotations, or any bounded chart should therefore enforce a boundary-normal condition or use a barrier/projected flow rather than relying on strong convexity alone. The sharp engineering prediction is that unconstrained updates leave the chart at a measurable step size, while a radial barrier or projection eliminates outward boundary velocity.