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arxiv: 2605.03484 · v1 · submitted 2026-05-05 · 🧮 math.OC · math.MG

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Quantitative Convergence of Proximal Splitting Iterations in Uniformly Convex Metric Spaces

D. Russell Luke , Mahshid Mirhashemi
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Pith reviewed 2026-05-07 15:33 UTC · model grok-4.3

classification 🧮 math.OC math.MG
keywords proximal splittingquantitative convergenceuniformly convex metric spacesHadamard spacesFréchet meansproximal operatorsoptimization on manifoldsCAT(kappa) spaces
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The pith

Proximal splitting algorithms converge at explicit rates in p-uniformly convex metric spaces even without common minima or vanishing step sizes.

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

This paper gives sufficient conditions for the iterates of proximal splitting algorithms to converge quantitatively when minimizing the sum of functions in a metric space. A sympathetic reader would care because these conditions allow explicit convergence rates even if the individual functions do not share a minimizer and without forcing the proximal parameters to go to zero, which is often impractical. The setting is general p-uniformly convex spaces with curvature bounded from above, with a simpler corollary for Hadamard spaces. The approach is demonstrated on the computation of Fréchet means for positive semidefinite matrices under the affine-invariant metric and on the sphere under the geodesic metric.

Core claim

We provide sufficient conditions for quantitative convergence of the iterates of proximal splitting algorithms for minimizing a sum of functions on a metric space. The theory does not assume that the functions have common minima, nor does it require vanishing proximal parameters or step sizes. Our results are stated for general p-uniformly convex spaces with curvature bounded above, and a corollary specializes the main theorem to Hadamard spaces, where many assumptions for the more general setting can be dropped. The theory is demonstrated with computation of Fréchet means in the space of SPD matrices with the affine invariant metric and the sphere with the usual geodesic metric.

What carries the argument

The proximal splitting iteration in p-uniformly convex metric spaces with curvature bounded above, where the proximal operators satisfy conditions that yield explicit convergence rates for the generated sequence.

Load-bearing premise

The metric space must be p-uniformly convex with curvature bounded above, and the proximal operators must satisfy the sufficient conditions for the quantitative rates.

What would settle it

A concrete computation or counterexample in a space that fails to be p-uniformly convex with bounded curvature, where the splitting iterates either diverge or converge without matching the predicted quantitative rate.

Figures

Figures reproduced from arXiv: 2605.03484 by D. Russell Luke, Mahshid Mirhashemi.

Figure 1
Figure 1. Figure 1: Convergence of Algorithm 1 for different damping parameters τ for computatoin of the Fr´echet mean on two data sets of size 20. Left: iteration on the Hadamard manifold (S 3 ++, d). Right: iteration on the sphere S 2 with the spherical metric. In each case, the plot shows the residual d(x k , xk+1) versus the iteration index k on a logarithmic scale. positive diagonal entries and set A = LL⊤. This yields s… view at source ↗
read the original abstract

We provide sufficient conditions for quantitative convergence of the iterates of proximal splitting algorithms for minimizing a sum of functions on a metric space. The theory does not assume that the functions have common minima, nor does it require vanishing proximal parameters or step sizes. Our results are stated for general $p$-uniformly convex spaces with curvature bounded above, and a corollary specializes the main theorem to Hadamard spaces, where many assumptions for the more general setting can be dropped. The theory is demonstrated with computation of Fr\'echet means in the space of SPD matrices with the affine invariant metric (a Hadamard space) and the sphere with the usual geodesic metric (a CAT($\kappa$) metric space).

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

1 major / 2 minor

Summary. The paper provides sufficient conditions for quantitative convergence of proximal splitting iterations minimizing a sum of functions on p-uniformly convex metric spaces with curvature bounded above. It does not require the functions to share minima or vanishing proximal parameters/step sizes. A corollary specializes to Hadamard spaces with relaxed assumptions. The theory is illustrated by Fréchet mean computations on SPD matrices (Hadamard) and the sphere (CAT(κ)).

Significance. If the quantitative rates hold globally, the results would meaningfully extend explicit convergence analysis beyond Hadamard spaces to positively curved manifolds, providing non-asymptotic bounds useful for manifold optimization without common-minima assumptions.

major comments (1)
  1. [§3] §3 (main theorem and its proof): the derivation of the explicit contraction (3.5) or (3.7) invokes the modulus of p-uniform convexity globally on the orbit. In CAT(κ) spaces with κ>0 this modulus is known to be local (controlling only distances below a κ-dependent threshold, typically <π/√κ). Without an auxiliary diameter bound on the iterates, the claimed quantitative guarantee does not extend to arbitrary initial data in the sphere demonstration of §5.2.
minor comments (2)
  1. The abstract and introduction should explicitly name the proximal splitting schemes (e.g., forward-backward, Douglas-Rachford) to which the sufficient conditions apply.
  2. Notation for the curvature bound and the precise definition of the proximal operator in the general p-uniformly convex setting could be clarified with a short remark on how it reduces to the Hadamard case.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading and for highlighting this important technical point on the locality of the p-uniform convexity modulus. We address the comment below.

read point-by-point responses

  1. Referee: [§3] §3 (main theorem and its proof): the derivation of the explicit contraction (3.5) or (3.7) invokes the modulus of p-uniform convexity globally on the orbit. In CAT(κ) spaces with κ>0 this modulus is known to be local (controlling only distances below a κ-dependent threshold, typically <π/√κ). Without an auxiliary diameter bound on the iterates, the claimed quantitative guarantee does not extend to arbitrary initial data in the sphere demonstration of §5.2.

    Authors: We agree that the modulus of p-uniform convexity is local in CAT(κ) spaces with κ > 0 and applies only for distances strictly less than π/√κ. The proof of the main contraction (3.5) in Theorem 3.1 applies the modulus to pairs of points along the orbit, which implicitly requires that all relevant distances remain below this threshold. In the Hadamard-space corollary the modulus is global, so the issue does not arise. For the sphere example (CAT(1)) in §5.2 the quantitative bound therefore holds only when the orbit diameter stays below π; this is typically satisfied for Fréchet-mean computations when the data lie in an open hemisphere, but the manuscript does not state an explicit diameter restriction on the initial point. We will add a clarifying remark after Theorem 3.1 and a short paragraph in §5.2 noting the locality condition and the sufficient assumption that the initial iterate lies in a ball of radius < π/2 (ensuring the orbit remains inside the region of applicability). With this addition the claimed rates become rigorous under the stated hypotheses. revision: partial

Circularity Check

0 steps flagged

No significant circularity; derivation self-contained from geometric assumptions

full rationale

The paper derives quantitative convergence rates for proximal splitting iterations from the stated geometric properties of p-uniformly convex metric spaces with curvature bounded above. The main result supplies sufficient conditions without reducing claimed rates to fitted parameters, self-definitions, or load-bearing self-citations. The Hadamard-space corollary drops assumptions cleanly, and the SPD-matrix and sphere demonstrations apply the conditions to concrete proximal operators without circular reduction of the contraction estimates to the inputs. No ansatz smuggling or renaming of known results appears in the derivation chain.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claim rests on the geometric assumptions of the ambient space and the existence of proximal operators; no free parameters or invented entities are introduced in the abstract.

axioms (2)
  • domain assumption The metric space is p-uniformly convex with curvature bounded above
    This is the general setting stated for the main theorem.
  • domain assumption Proximal operators exist and satisfy the conditions needed for the splitting scheme
    Required for the algorithms to be well-defined and for the convergence statements to apply.

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