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Forward-backward splitting in bilaterally bounded Alexandrov spaces

Heikki von Koch, Tuomo Valkonen

TL;DR

The paper extends proximal gradient methods to bilaterally bounded Alexandrov spaces, where curvature is bounded above and below, enabling both gradient and proximal steps on non-Riemannian manifolds. It defines a proximal gradient map using the exponential (and gradient) maps, derives a key descent inequality that handles the composite objective $H=F+G$ under curvature constraints, and proves convergence of the forward-backward iteration under suitable step-size and diameter conditions. The analysis yields explicit constants tied to curvature bounds, and the results are demonstrated numerically on simple geometries such as the cube surface and a capped cylinder, including explicit formulas for gradients and proximal updates in these settings. This work broadens optimization on manifolds beyond Hadamard spaces, with potential applications to optimization on geometrically irregular surfaces and more general metric spaces.

Abstract

With the goal of solving optimisation problems on non-Riemannian manifolds, such as geometrical surfaces with sharp edges, we develop and prove the convergence of a forward-backward method in Alexandrov spaces with curvature bounded both from above and from below. This bilateral boundedness is crucial for the availability of both the gradient and proximal steps, instead of just one or the other. We numerically demonstrate the behaviour of the proposed method on simple geometrical surfaces in $\mathbb{R}^3$.

Forward-backward splitting in bilaterally bounded Alexandrov spaces

TL;DR

The paper extends proximal gradient methods to bilaterally bounded Alexandrov spaces, where curvature is bounded above and below, enabling both gradient and proximal steps on non-Riemannian manifolds. It defines a proximal gradient map using the exponential (and gradient) maps, derives a key descent inequality that handles the composite objective under curvature constraints, and proves convergence of the forward-backward iteration under suitable step-size and diameter conditions. The analysis yields explicit constants tied to curvature bounds, and the results are demonstrated numerically on simple geometries such as the cube surface and a capped cylinder, including explicit formulas for gradients and proximal updates in these settings. This work broadens optimization on manifolds beyond Hadamard spaces, with potential applications to optimization on geometrically irregular surfaces and more general metric spaces.

Abstract

With the goal of solving optimisation problems on non-Riemannian manifolds, such as geometrical surfaces with sharp edges, we develop and prove the convergence of a forward-backward method in Alexandrov spaces with curvature bounded both from above and from below. This bilateral boundedness is crucial for the availability of both the gradient and proximal steps, instead of just one or the other. We numerically demonstrate the behaviour of the proposed method on simple geometrical surfaces in .

Paper Structure

This paper contains 15 sections, 29 theorems, 120 equations, 4 figures, 1 table.

Table of Contents

  1. Introduction
  2. Alexandrov spaces
  3. Convex analysis in Alexandrov spaces
  4. Trigonometric bounds
  5. Proximal and gradient descent maps
  6. The proximal map
  7. Gradient descent map
  8. Proximal gradient map
  9. The key estimate
  10. Forward-backward method
  11. Convergence analysis
  12. Examples
  13. Auxiliary lemma
  14. Cube
  15. Cylinder

Key Result

Theorem 2.6

Let $(X,d)$ be an Alexandrov space with curvature bounded from above or below by $\kappa$, and assume that $X$ is locally compact when the curvature is bounded from below. For any point $z \in X$ and geodesic $\gamma(t) = x \#_t y$ with $x, y \in X$, and $0 < d(x,z) < \pi / \sqrt{\kappa}$ if $\kappa where $d_z: X \to \mathbb{R} : a \to d(z,a)$ and $\theta_{\min}$ is the minimum of angles between $

Figures (4)

  • Figure 1: A unit cube with labeled faces and their local coordinate systems.
  • Figure 2: Two copies of an unfolding of the cube with the labels of the faces marked on their upper left corners. Included are three colour-coded paths from face $1$ to face $2$, with different starting points but same endpoints, in the coordinate system of face $1$. The paths on the left are all geodesics, whereas the dashed red path on the right is not, as it goes through the vertex $(1,0)_1$. Note that the two zigzag edges between faces $2$ are not real but merely artefacts of the drawing.
  • Figure 3: Cylinder tangents and geodesics. In the flattened 2D view of (\ref{['fig:cyltangent:2d']}), geodesics are straight lines; for any two points on the geodesic, the tangent vector given by the logarithmic map is parallel to this line. In the 3D view of (\ref{['fig:cyltangent:3d']}), the same geodesic is a helical path on the side, and again a straight line on the top.
  • Figure 4: Numerical illustrations. The surface shading indicates function value. Blue dots indicate data points for squared distance terms on exposed faces, while dimmer dots indicate data points on hidden faces. The green circle indicates the “origin” data point for the unsquared distance term. Algorithm iterates and connecting lines are shown for three different starting points.

Theorems & Definitions (75)

  • Definition 2.1: Alexandrov spaces
  • Example 2.2
  • Definition 2.3: The tangent cone
  • Definition 2.4: petruninfoundations
  • Remark 2.5
  • Theorem 2.6: First variation formula
  • Remark 2.7
  • Theorem 2.8: berestovskii
  • Remark 2.9
  • Definition 2.10
  • ...and 65 more