Convergence of inexact descent methods for nonconvex optimization on Riemannian manifolds
Provides theoretical convergence guarantees for optimization algorithms on manifolds, relevant for nonconvex problems in machine learning and signal processing.
The paper proves convergence of inexact descent methods (proximal point and steepest descent) on Riemannian manifolds for nonconvex, nonsmooth functions satisfying the Kurdyka-Lojasiewicz inequality, without restrictive curvature assumptions.
In this paper we present an abstract convergence analysis of inexact descent methods in Riemannian context for functions satisfying Kurdyka-Lojasiewicz inequality. In particular, without any restrictive assumption about the sign of the sectional curvature of the manifold, we obtain full convergence of a bounded sequence generated by the proximal point method, in the case that the objective function is nonsmooth and nonconvex, and the subproblems are determined by a quasi distance which does not necessarily coincide with the Riemannian distance. Moreover, if the objective function is $C^1$ with $L$-Lipschitz gradient, not necessarily convex, but satisfying Kurdyka-Lojasiewicz inequality, full convergence of a bounded sequence generated by the steepest descent method is obtained.