Convex analysis on Hadamard spaces and scaling problems

TL;DR

This paper develops a convex analysis framework for geodesically convex optimization on Hadamard spaces, extending recession function concepts to analyze boundedness and unboundedness, with applications to operator scaling and group orbit optimization.

Contribution

It introduces an extension of the recession function via convex analysis on Hadamard spaces, providing tools for unboundedness analysis in geodesic convex optimization.

Findings

Extended recession function concept to Hadamard spaces.

Provided convex analysis foundation for unboundedness determination.

Applied theory to operator scaling and group orbit optimization.

Abstract

In this paper, we address the bounded/unbounded determination of geodesically convex optimization on Hadamard spaces. In Euclidean convex optimization, the recession function is a basic tool to study the unboundedness, and provides the domain of the Legendre-Fenchel conjugate of the objective function. In a Hadamard space, the asymptotic slope function (Kapovich, Leeb, and Millson 2009), which is a function on the boundary at infinity, plays a role of the recession function. We extend this notion by means of convex analysis and optimization, and develop a convex analysis foundation for the unbounded determination of geodesically convex optimization on Hadamard spaces, particularly on symmetric spaces of nonpositive curvature. We explain how our developed theory is applied to operator scaling and related optimization on group orbits, which are our motivation.