Gradient Descent for Unbounded Convex Functions on Hadamard Manifolds and Its Applications to Scaling Problems

In this paper, we study the asymptotic behavior of continuous- and discrete-time gradient flows of a “lower-unbounded” convex function f on a Hadamard manifold M, particularly their convergence properties to the boundary [Formula: see text] at infinity of M. We establish a duality theorem that the infimum of the gradient-norm [Formula: see text] of f over M is equal to the supremum of the negative of the recession function [Formula: see text] of f over the boundary [Formula: see text], provided the infimum is positive. Further, the infimum and the supremum are obtained by the limit of the gradient flow of f. Our results feature convex optimization ingredients of the moment-weight inequality for reductive group actions, and are applied to noncommutative optimization. We show that gradient descent of the Kempf-Ness function for an unstable orbit converges to a destabilizing 1-parameter subgroup in the Hilbert-Mumford criterion, and the associated moment-map sequence converges to the minimum-norm point of the moment polytope. We show further refinements for operator scaling—the left-right action on a matrix tuple [Formula: see text]. We characterize the gradient-flow limit of operator scaling by a vector-space generalization of the classical Dulmage-Mendelsohn decomposition of a bipartite graph. For a special case of [Formula: see text], we reveal that the limit determines the Kronecker canonical form of a matrix pencil [Formula: see text]. Funding: H. Hirai was supported by the Japan Society for the Promotion of Science (KAKENHI [Grants JP21K19759 and JP24K21315]). K. Sakabe was supported by Deutsche Forschungsgemeinschaft (DFG, German Research Foundation [Grant 556164098]) and the European Research Council (ERC Starting Grant SYMOPTIC [Grant 101040907]).