Metric-Free Riemannian Optimization
Riemannian optimization provides a powerful framework for constrained optimization by incorporating problem-specific structure directly into the geometry of the search space. In many applications, however, the explicit evaluation or application of the Riemannian metric can be computationally expensive or numerically unstable, limiting the practical efficiency of otherwise well-founded algorithms. Motivated by such settings, this work investigates to what extent classical Riemannian optimization algorithms can be reformulated without explicitly applying the metric. We show that many first-order components of Riemannian optimization only rely on the differential of the objective function and access to the Riemannian gradient, but not on explicit metric application. Based on this observation, we develop metric-free formulations and generalize optimization approaches to Finsler and Banach manifolds. Numerical experiments demonstrate that the proposed metric-free strategies retain the effectiveness of their metric-dependent counterparts while significantly reducing computational overhead. These results highlight that a substantial portion of Riemannian optimization can be carried out independently of explicit metric application, broadening its applicability to problems with expensive or implicitly defined metrics.