A Riemannian perspective on generalized smoothness

In this work we develop Riemannian optimization formulations for optimization problems posed on Euclidean spaces, and that may not satisfy classical smoothness criteria. We introduce a family of metrics that can confirm global Riemannian smoothness in function classes including certain neural-network training problems, enabling convergence guarantees for Riemannian optimization schemes applied to these objectives. The metrics are conformal scalings and warped products defined using radial scaling functions, and we further derive low-dimensional geodesic equations for these metrics, which can lead to efficient implementations. Finally, we consider connections with $(L_0,L_1)$-smoothness and $\ell$-smoothness criteria.