Stochastic Primal-Dual Method on Riemannian Manifolds with Bounded Sectional Curvature

Abstract

We study a stochastic primal-dual method for constrained optimization over Riemannian manifolds with bounded sectional curvature. We prove non-asymptotic convergence to the optimal objective value. More precisely, for the class of hyperbolic manifolds, we establish a convergence rate that is related to the sectional curvature lower bound. To prove a convergence rate in terms of sectional curvature for the elliptic manifolds, we leverage Toponogov's comparison theorem. In addition, we provide convergence analysis for the asymptotically elliptic manifolds, where the sectional curvature at each given point on manifold is locally bounded from below by the distance function. We demonstrate the performance of the primal-dual algorithm on the sphere for the non-negative principle component analysis (PCA). In particular, under the non-negativity constraint on the principle component and for the symmetric spiked covariance model, we empirically show that the primal-dual approach outperforms the spectral method. We also examine the performance of the primal-dual method for the anchored synchronization from partial noisy measurements of relative rotations on the Lie group SO(3). Lastly, we show that the primal-dual algorithm can be applied to the weighted MAX-CUT problem under constraints on the admissible cut. Specifically, we propose different approximation algorithms for the weighted MAX-CUT problem based on optimizing a function on the manifold of direct products of the unit spheres as well as the manifold of direct products of the rotation groups.