On projection mappings and the gradient projection method on hyperbolic space forms

TL;DR

This work addresses constrained optimization on $\\kappa$-hyperbolic space forms by exploiting intrinsic $\\kappa$-projections onto $\\kappa$-hyperbolically convex sets and clarifying their relationship to Lorentz and Euclidean projections. It establishes existence, continuity, and computability of the intrinsic projection for broad classes of sets, and provides explicit projection formulas for select convex sets via reductions to cone projections in the Lorentz model. The authors introduce two intrinsic gradient projection methods—one with constant stepsize and one with backtracking stepsize—proving that accumulation points are stationary and deriving iteration-complexity bounds without requiring convexity or compactness, and they apply these results to the constrained Fermat-Weber problem with convergence to a unique center of mass. Numerical experiments on the constrained Riemannian center of mass validate the theory and demonstrate the practical efficiency of the proposed methods compared to ALM and EPM, especially in higher dimensions.

Abstract

This paper presents several new properties of the intrinsic $κ$-projection into $κ$-hyperbolically convex sets of $κ$-hyperbolic space forms, along with closed-form formulas for the intrinsic $κ$-projection into specific $κ$-hyperbolically convex sets. It also discusses the relationship between the intrinsic $κ$-projection, the Euclidean orthogonal projection, and the Lorentz projection. These properties lay the groundwork for analyzing the gradient projection method and hold importance in their own right. Additionally, new properties of the gradient projection method to solve constrained optimization problems in $κ$-hyperbolic space forms are established, considering both constant and backtracking step sizes in the analysis. It is shown that every accumulation point of the sequence generated by the method for both step sizes is a stationary point for the given problem. Additionally, an iteration complexity bound is provided that upper bounds the number of iterations needed to achieve a suitable measure of stationarity for both step sizes. Finally, the properties of the constrained Fermat-Weber problem are explored, demonstrating that the sequence generated by the gradient projection method converges to its unique solution. Numerical experiments on solving the Fermat-Weber problem are presented, illustrating the theoretical findings and demonstrating the effectiveness of the proposed methods.

Figures (3)

• Figure 1: Illustration a section of the Poincaré ball, the majority of the sampled points outside the constrained set illustrated as an orange dashed circle. The Riemannian center of mass lies outside the constrained set. The projected mean and all minimisers lie very close together, while the projected mean is still higher in cost and a reasonable distance away.
• Figure 2: Objective versus iteration plot for different solvers on the 2D constrained mean.
• Figure 3: A comparison of the exact penalty method (EPM), augmented Lagrangian mathod (ALM), and the projected gradient for the constrained Riemannian center of mass on $\mathbb H_1^d$, $d=2,\ldots,200$: left: the number of iterations per experiment (right) the runtime per experimant

Theorems & Definitions (58)

• Remark 1
• Remark 2
• Lemma 1
• Lemma 2
• Definition 1
• Lemma 3
• Lemma 4
• Proposition 5
• Remark 3
• Proposition 6