Gradient descent for unbounded convex functions on Hadamard manifolds and its applications to scaling problems

TL;DR

The paper investigates gradient flows and gradient-descent dynamics for lower-unbounded geodesically convex functions on Hadamard manifolds, establishing a strong duality between the infimum of the gradient norm and the recession function over the boundary at infinity. It proves that, under positive minimum gradient-norm, the gradient flow and its discrete counterpart converge to a unique boundary direction that minimizes the recession function, providing a certificate of unboundedness. The framework is then applied to norm-minimization in reductive group actions (Kempf-Ness) and to operator scaling, uncovering a deep link with DM-decompositions and yielding insights into matrix scaling, moment maps, and Kronecker form classifications. These results unify continuous and discrete optimization on nonpositive-curvature manifolds with invariant-theoretic and representation-theoretic perspectives, offering algorithmic certificates and asymptotic descriptions for unscalability and structural decompositions. The work also connects to geometric programming and convex analysis, illustrating how gradient-based methods reveal intrinsic boundary structures such as the DM- and Kronecker-type decompositions in noncommutative optimization Problems.

Abstract

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 Bürgisser 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 $s A_1+A_2$.

Figures (1)

• Figure 1: $\mathop{\rm Conv} \phi({\cal S}_A)$ in $(y,x)$-plane (left) and a DM-decomposition of $A$ (right). The slope $n_{\alpha}/m_{\alpha}$ is increasing by the convexity of $\mathop{\rm Conv} \phi({\cal S}_A)$.

Theorems & Definitions (74)

• Proposition 2.1: KLM2009JDG; see also Hirai_Hadamard2022
• Lemma 2.2: Weak duality
• proof
• Theorem 3.1
• Lemma 3.2
• proof
• proof : Proof of Theorem \ref{['thm:continuous_main']}
• Corollary 3.3
• Proposition 3.4
• proof