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 $M^{\infty}$ at infinity of $M$. We establish a duality theorem that the infimum of the gradient-norm $\|\nabla f(x)\|$ of $f$ over $M$ is equal to the supremum of the negative of the recession function $f^{\infty}$ of $f$ over the boundary $M^{\infty}$, 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 by Georgoulas, Robbin, and Salamon, and are applied to noncommutative optimization by B\"urgisser et al. FOCS 2019. 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 $A= (A_1,A_2,\ldots,A_N)$. 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 $s A_1+A_2$.