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arxiv: 2605.03815 · v1 · submitted 2026-05-05 · 🧮 math.OC

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Ball-proximal point method on a Hadamard Manifolds

F. Babu , O. P. Ferreira , L. F. Prudente , Jen-Chih Yao , Xiaopeng Zhao

Authors on Pith no claims yet

Pith reviewed 2026-05-07 15:10 UTC · model grok-4.3

classification 🧮 math.OC
keywords ball-proximal point methodHadamard manifoldgeodesically convex functionRiemannian optimizationproximal algorithmquasi-Fejér monotonicityconvergence analysis
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The pith

A Riemannian ball-proximal point method converges to minimizers of geodesically convex functions on Hadamard manifolds when the sum of radii diverges.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper introduces a ball-proximal point method adapted to Hadamard manifolds for minimizing proper lower semicontinuous geodesically convex functions. Each iteration minimizes the objective over a metric ball around the current iterate, using the Riemannian broximal map. The method establishes quasi-Fejér monotonicity, a product-form linear decay of function values until the solution set is hit, and finite termination for constant positive radii. When the sum of radii diverges, objective values reach the optimum and iterates converge to a minimizer if the solution set is nonempty.

Core claim

The Riemannian ball-proximal point method generates sequences that are quasi-Fejér monotone with respect to the solution set, exhibit a product-form linear decay of objective values up to the hitting time of the solution set, and converge to a minimizer whenever the sum of radii diverges and the solution set is nonempty. These properties follow from existence and uniqueness of broximal points together with a KKT-type characterization on Hadamard manifolds.

What carries the argument

The Riemannian broximal map, which returns the unique minimizer of the objective inside a closed metric ball centered at the current iterate and is characterized by a nonnegative scalar multiplier and an element of the Riemannian subdifferential.

If this is right

  • Squared distance to the solution set decreases strictly whenever the current ball does not contain a minimizer.
  • The method terminates in finitely many steps when radii are held constant and positive.
  • Nonasymptotic bounds hold on the norms of subgradients and on function values, with a linear rate when radii are constant.
  • Objective values converge to the optimal value whenever the sum of radii diverges.
  • The full sequence of iterates converges to a minimizer when the solution set is nonempty and the sum of radii diverges.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The ball-proximal construction may serve as a template for other proximal-type algorithms on manifolds where geodesic convexity is available but full proximal maps are expensive to compute.
  • Adaptive radius schedules that depend on observed function decrease could be tested numerically to improve practical performance while preserving the divergence condition.
  • The same quasi-Fejér framework might extend to inexact ball minimizations, provided the inexactness is controlled in a summable way.

Load-bearing premise

The objective must be proper, lower semicontinuous, and geodesically convex on a complete simply-connected manifold of non-positive sectional curvature.

What would settle it

A concrete sequence of iterates on hyperbolic space where the sum of radii diverges yet the distance to the solution set remains bounded away from zero would refute the asymptotic convergence claim.

Figures

Figures reproduced from arXiv: 2605.03815 by F. Babu, Jen-Chih Yao, L. F. Prudente, O. P. Ferreira, Xiaopeng Zhao.

Figure 1
Figure 1. Figure 1: Number of function evaluations for representative parameter choices: Broximal-A ( view at source ↗
Figure 2
Figure 2. Figure 2: Illustrative pointwise paths on a two-dimensional strongly convex quadratic problem. view at source ↗
read the original abstract

We consider the problem of minimizing a proper, lower semicontinuous, geodesically convex function on a Hadamard manifold. Building on ball-proximal (broximal) ideas in the Euclidean setting, viewed as an abstract proximal-type algorithm, we propose and analyze a Riemannian ball-proximal point method (RB-PPM) whose basic step consists of minimizing the objective function over a metric ball centred at the current iterate. We first introduce the Riemannian broximal map, prove existence and uniqueness of broximal points on Hadamard manifolds, and derive a KKT-type characterization involving a scalar parameter and the Riemannian subdifferential. We then show that RB-PPM enjoys a strict decrease of the squared distance to the solution set whenever the current ball does not contain a minimizer. This leads to quasi-Fej\'er monotonicity, finite termination for constant radii, and a product-form linear decay of the objective values up to the hitting time of the solution set. We also obtain nonasymptotic complexity bounds for the norms of suitable subgradients and for the function values, including a linear rate in the number of iterations under constant radii. Finally, we establish an asymptotic dichotomy, if the sum of the radii diverges, then the objective values converge to the optimal value, and, when the solution set is nonempty, the entire sequence of iterates converges to a minimizer. The resulting scheme provides a geometry-aware, ball-based analog of classical Riemannian proximal point methods.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

0 major / 4 minor

Summary. The manuscript proposes and analyzes a Riemannian ball-proximal point method (RB-PPM) for minimizing a proper, lower semicontinuous, geodesically convex function on a Hadamard manifold. The basic iteration minimizes the objective over a metric ball centered at the current point. The authors introduce the Riemannian broximal map, establish its existence and uniqueness, derive a KKT-type characterization using the Riemannian subdifferential, prove strict decrease of the squared distance to the solution set (when the ball does not contain a minimizer), quasi-Fejér monotonicity, finite termination under constant radii, product-form linear decay of function values, nonasymptotic complexity bounds on subgradient norms and function values (including linear rates), and an asymptotic dichotomy: divergence of the radius sum implies convergence of function values to the optimum and, when the solution set is nonempty, convergence of the iterates to a minimizer.

Significance. If the central claims hold, the work supplies a geometry-aware ball-based proximal-type algorithm on Hadamard manifolds that complements existing Riemannian proximal point methods. The finite-termination result for constant radii, the explicit product-form linear decay, and the nonasymptotic bounds are concrete strengths that could be useful for both theory and implementation. The reliance on standard geodesic convexity together with Hadamard geometry (non-positive curvature, completeness) to obtain convexity of the squared distance function is a clean extension of the Euclidean ball-proximal framework.

minor comments (4)
  1. In the statement of the KKT-type characterization for the broximal map, the scalar parameter appears without an explicit range or sign condition; adding a short remark on its sign would clarify the subdifferential inclusion.
  2. The proof of quasi-Fejér monotonicity (likely in the section following the definition of the broximal map) invokes the triangle inequality for the Riemannian distance; a one-line reference to the relevant Hadamard-manifold inequality would help readers.
  3. The nonasymptotic bound on subgradient norms is stated in terms of the radii sequence; an explicit dependence on the initial distance to the solution set should be written out for completeness.
  4. A brief comparison paragraph with the classical Riemannian proximal point method (e.g., rate differences under constant versus diminishing radii) would strengthen the introduction.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the careful reading of the manuscript and the positive assessment, including the accurate summary of our contributions on the Riemannian ball-proximal point method, its properties such as quasi-Fejér monotonicity, finite termination under constant radii, and the asymptotic convergence results under diverging radius sums. The recommendation for minor revision is appreciated. No specific major comments were raised in the report.

Circularity Check

0 steps flagged

No significant circularity in derivation chain

full rationale

The paper defines the Riemannian broximal map as the argmin of the objective over a metric ball centered at the current point, then derives existence/uniqueness, KKT characterization, strict decrease of squared distance to the solution set, quasi-Fejér monotonicity, finite termination, nonasymptotic bounds, and the asymptotic dichotomy directly from the standing assumptions of proper lsc geodesic convexity on a Hadamard manifold. These steps rely on standard properties of the Riemannian distance function, geodesic convexity, and the non-positive curvature geometry; none reduce by construction to fitted parameters, self-referential equations, or load-bearing self-citations whose validity is presupposed by the present work. The central claims remain independent of the target results and are externally falsifiable via the manifold axioms.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 1 invented entities

The central claims rest on standard Riemannian geometry assumptions and the introduction of a new broximal map; no free parameters are mentioned.

axioms (2)
  • domain assumption Hadamard manifold: complete, simply connected Riemannian manifold with non-positive sectional curvature
    Invoked for existence and uniqueness of broximal points and for geodesic convexity properties.
  • domain assumption Objective function is proper, lower semicontinuous, and geodesically convex
    Stated in the problem formulation and used throughout the analysis.
invented entities (1)
  • Riemannian broximal map no independent evidence
    purpose: Defines the ball-proximal point as the minimizer of the objective over a metric ball
    New concept introduced to generalize the Euclidean ball-proximal operator; no independent evidence provided beyond the paper's existence proof.

pith-pipeline@v0.9.0 · 5583 in / 1473 out tokens · 71317 ms · 2026-05-07T15:10:28.372387+00:00 · methodology

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Reference graph

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