Gradient Descent for Unbounded Convex Functions on Hadamard Manifolds and Its Applications to Scaling Problems
成果类型：
Article; Early Access
署名作者：
Hirai, Hiroshi; Sakabe, Keiya
署名单位：
Nagoya University; University of Munich
刊物名称：
MATHEMATICS OF OPERATIONS RESEARCH
ISSN/ISSBN：
0364-765X; 1526-5471
DOI：
10.1287/moor.2025.0939
发表日期：
2026-04-24
关键词：
Hadamard manifold
geodesically convex optimization
gradient flow
matrix scaling
geometric programming
Hilbert-Mumford criterion
Kempf-Ness theorem
moment polytope
operator scaling
Dulmage-Mendelsohn decomposition
optimization
FLOWS
inequalities
computation
matrices
SPACES
rank
form
摘要：
In this paper, we study the asymptotic behavior of continuous- and discretetime gradient flows of a lower-unbounded convex function f on a Hadamard manifold M, particularly their convergence properties to the boundary M infinity at infinity of M. We establish a duality theorem that the infimum of the gradient-norm & Vert; del f(x) & Vert; of f over M is equal to the supremum of the negative of the recession function f infinity of f over the boundary M infinity, 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 A = (A1,A2,...,AN). 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 N = 2, we reveal that the limit determines the Kronecker canonical form of a matrix pencil sA1 + A2.
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