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On the Rate of Asymptotic Regularity of Iterative Methods for Nonexpansive Mappings in CAT(0) Spaces and Hyperbolic Optimization

T0 review · 3 major / 3 minor · reviewed 2026-08-04 · deepseek-v4-flash

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Pith's one-line read This paper claims that the classical Euclidean asymptotic-regularity bounds for the Krasnoselskii–Mann and Halpern fixed-point iterations transfer verbatim to complete CAT(0) spaces, and it introduces a hyperbolic optimizer built from the H

desk verdict The KM-rate theorem in CAT(0) looks correct, but the viscosity/Halpern O(1/k) proof rests on a false lemma as stated. read the letter →

arxiv 2510.25363 v3 pith:REAHWX7O submitted 2025-10-29 math.OC math.FA

Katherine Rossella Foglia , Vittorio Colao This is my paper · ORCID
classification math.OCmath.FA MSC 47H0947H10
keywords CAT(0)spacesKrasnosel'skii–ManniterationHalpernnonexpansivemappingsasymptoticregularityconvergencerateHadamardhyperbolicoptimization
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

Two classical fixed-point algorithms—the Krasnosel'skii–Mann (KM) iteration and the Halpern (viscosity) iteration—are known to converge to fixed points of nonexpansive maps at explicit rates in Euclidean and Banach spaces. This paper attempts to show that those exact rates survive in complete CAT(0) spaces, the geodesic spaces of non-positive curvature that include hyperbolic geometry. If correct, the KM iteration keeps its O(1/√Σλ_i(1−λ_i)) bound and the Halpern iteration keeps its faster O(1/k) bound in these curved spaces. The paper then packages this into a 'Hyperbolic HalpernGD' optimizer, replacing the Euclidean gradient step by the proximal (resolvent) map, whose fixed points are exactly the minimizers of a convex objective. The broader stake is that acceleration by anchoring—not just convergence—carries over to the geometry used in hyperbolic deep learning.

What carries the argument

The carrying mechanism is the set of probability-like weights π_n^k = λ_k ∏_{j=k+1}^n (1−λ_j). These weights turn each KM iterate into a convex combination of past displacement terms d(T x_{j−1}, ·), and in CAT(0) spaces the metric convexity inequality supplies the same triangle-type bounds that in Banach spaces come from linearity. The recursion c_{m,n} that governs the distance between iterates is defined from these same weights. For the Halpern side, the engine is the scalar recursion a_{k+1} ≤ (1−γ b_{k+1})a_k + (b_k − b_{k+1})c_k with b_k = 2/((1−β)k), which converts a nonincreasing step into an O(1/k) displacement bound.

What would settle it

Check the disputed equality in the proof of Proposition 4.5 at m=1, n=2: with any λ_1,λ_2∈(0,1), the proof bounds the j=0 term by a sum over k=1,2, but c_{1,2} only includes k=2. The extra term λ_1(1−λ_1)(1−λ_2) is strictly positive, so the displayed identity is false; whether Theorem 4.1's final bound still holds then has to be re-established, or the definition of c_{m,n} must be widened and the π factor recomputed.

Watch

Extended reading notes

Core claim

The load-bearing results are Theorem 4.1 and Theorem 4.11. Theorem 4.1 states that in a complete CAT(0) space, the KM iteration satisfies d(x_n,T x_n) ≤ diam(K)√(π/Σ_{i=1}^n λ_i(1−λ_i)), reproducing the classical Euclidean bound. Theorem 4.11 states that the viscosity-type Halpern iteration, with step α_k = min{2/((1−β)k),1}, satisfies d(T x_{k−1}, x_{k−1}) ≤ 2C(J+2)/((1−β)k), i.e., O(1/k). The proof machinery, imported from the linear setting, works because CAT(0) spaces satisfy the metric convexity inequality used at each step, and because the coefficient weights are purely algebraic. The same fixed-point structure is then repurposed into a hyperbolic optimizer whose resolvent step is firm

Load-bearing premise

The KM rate theorem stands on a key coefficient lemma (Proposition 4.5), and in the proof as written one term is bounded using a wider index range than the definition of c_{m,n} allows, so the inequality d(x_n,x_m) ≤ c_{m,n} is not proven as stated.

Editorial extensions

If this is right

  • In complete CAT(0) spaces, the KM iteration's asymptotic regularity is bounded by diam(K)√(π/Σ_{i=1}^n λ_i(1−λ_i)), so any divergent series of λ_i(1−λ_i) gives O(1/√Σ...), exactly as in Euclidean spaces.
  • The viscosity/Halpern iteration with step α_k = min{2/((1−β)k),1} has d(T x_{k−1}, x_{k−1}) ≤ 2C(J+2)/((1−β)k), matching the Hilbert-space O(1/k) rate.
  • Halpern-type anchoring is therefore asymptotically faster than KM-type (gradient-descent-like) iteration in CAT(0) spaces, extending the Euclidean acceleration narrative to non-positively curved geometries.
  • The proposed Hyperbolic HalpernGD scheme, built from the proximal map, converges to a minimizer of a convex function in Hadamard spaces, providing a principled hyperbolic counterpart to the Euclidean anchored-gradient optimizer.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • Editorial inference: If the O(1/k) rate survives stochastic approximation, hyperbolic neural-network training could inherit the practical speed-ups attributed to Halpern-type anchoring, but only if the resolvent can be computed efficiently—something the paper leaves to future experiments.
  • Editorial inference: Because the proof only uses metric convexity, the same rate-transfer argument may extend to other geodesic settings, such as CAT(κ) spaces with κ≤0 or uniformly convex geodesic spaces, with diameter or radius conditions.
  • Editorial inference: The summation-index mismatch in Proposition 4.5 is localized; if it is a typo and the coefficient definition is widened to include the missing k=1..m terms, the π constant in Theorem 4.1 might change, so the final √π factor is the part to re-check.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

3 major / 3 minor

Summary. The paper studies the asymptotic regularity of the Krasnosel'skič–Mann (KM) and Halpern/viscosity iterations for nonexpansive mappings in complete CAT(0) spaces. It claims two main results: (i) Theorem 4.1, an O(1/√ n) bound on d(x_n,Tx_n) for the KM iteration (21), matching the Euclidean rate of Cominetti et al.; and (ii) Theorem 4.11, an O(1/k) bound on d(x_k,x_{k-1}) and d(Tx_{k-1},x_{k-1}) for the viscosity/Halpern iteration (31), matching the Euclidean rate of Sabach–Shtern. The paper also proposes a 'Hyperbolic HalpernGD' optimizer based on the proximal mapping in Hadamard spaces, motivated by an analogy to Euclidean HalpernSGD. The proofs are meant to transfer known linear-space arguments to the CAT(0) setting using metric convexity and nonexpansiveness.

Significance. If the two main theorems were correctly proven, the paper would establish that the classical Euclidean asymptotic-regularity rates for KM and Halpern iterations persist in complete CAT(0) spaces, a setting that includes hyperbolic space. That would be a useful and nontrivial contribution, since nonlinear geometry is known to sometimes worsen or complicate rates. The paper is also refreshingly self-contained: the transfer to CAT(0) spaces is explicit, the constants are explicit and depend only on the diameter (or on the contraction and initial distance), and no fitted parameters or numerical predictions are used to support the theory. The proposed Hyperbolic HalpernGD optimizer is conceptually reasonable, but it is presented as a theoretical proposal only, with no experiments; that is acceptable for an optimization-theory paper if the underlying rates are sound. The main issue is that two load-bearing proof steps are currently invalid as written, so the central claims are not yet established.

major comments (3)
  1. [Proposition 4.5] The proof of Proposition 4.5 has a summation-index mismatch that breaks the induction. The first term is bounded by ∑_{k=1}^n π^m_0 π^n_k c_{k-1}^{-1}, while c_{m,n} is defined with the inner sum over k=m+1..n. For m>0 the two sums differ by the nonnegative terms k=1..m, so the displayed equality '= c_{m,n}' does not follow. Changing the first term to k=m+1..n (or adjusting the definition of c_{m,n}) would close the induction, but as written Theorem 4.1, which rests on Proposition 4.5, is not proven.
  2. [Lemma 4.9 and Theorem 4.11] Lemma 4.9 is false as stated. For example, take b_k≡1, c_k≡0, γ=1, M=1, a_1=100; the recurrence hypothesis is satisfied but the conclusion would force a_1≤J=2. Moreover, even if the lemma were true with the stated direction, its application in Theorem 4.11 sets b_k=α_k=2/((1-β)k), so the conclusion a_k≤ MJ/b_k would give a_k≤ M J (1-β)k/2, which is O(k), not O(1/k). The line 'Therefore d(x_k,x_{k-1}) ≤ 2JC̄_x/((1-β)k)' does not follow from the stated lemma; a corrected lemma of the form a_k≤ MJ b_k or a direct induction is needed. This is a load-bearing gap for the O(1/k) viscosity/Halpern rate.
  3. [Theorem 4.11, second estimate] Even after repairing the first estimate, the derivation of d(Tx_{k-1},x_{k-1}) ≤ 2C̄_x(J+2)/((1-β)k) uses both the unproved first estimate and the bound d(f(x_{k-1}),Tx_{k-1}) ≤ 2C̄_x, which depends on the specific constant C̄_x from Lemma 4.8. The constants should be verified once the preliminary bound is fixed. As written, both displayed rates in Theorem 4.11 are unsupported.
minor comments (3)
  1. [General] Reference [57] is a duplicate of [53] (Sabach–Shtern); please consolidate. The abstract says 'chapter' where 'paper' is intended.
  2. [Remark 4.7] The 'probabilistic interpretation' of the weights π^n_k is invoked without a self-contained explanation; either give the argument or cite [49] explicitly for that step, since it is used to finish Theorem 4.1.
  3. [Section 5] The Hyperbolic HalpernGD optimizer is only a proposal: no implementation, experimental evaluation, or stochastic variant is provided. The statements in Section 6 that it 'should exhibit superior convergence speed' are expectations, not demonstrated results; please phrase them conditionally.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the KM and Halpern/viscosity rates are transferred from external linear-space proofs; the paper's self-citations are motivational and do not carry the proof load.

full rationale

The derivation chain is self-contained against external benchmarks. Theorem 4.1 is proved via Lemmas 4.2–4.5 and Remark 4.7, where the only imported estimate is the probabilistic bound of Cominetti [49], an external, parameter-free result; the CAT(0) geometry enters only through the convexity condition and nonexpansiveness, neither of which encodes the claimed rate. Theorem 4.11 similarly adapts the external Sabach–Shtern argument [53,57]; the constants C_x̄, J and M are defined from the data rather than fitted to the target inequality, and no 'predicted' quantity is constructed from the conclusion it is supposed to explain. The self-citations [1,2] appear only as motivation for HalpernSGD in the Euclidean setting and for expected empirical gains, and they do not justify any theorem here. Section 5 defines the Hyperbolic HalpernGD scheme as the Halpern iteration over the resolvent; this is an application of known Halpern convergence plus the standard fixed-point characterization of Jλ, not a derivation whose conclusion is assumed. The paper also explicitly concedes that part of the results were already investigated in the literature, which is a novelty limitation rather than circularity, and it flags that nonexpansivity of the forward operator in nonlinear spaces is still open. The reviewer-identified summation-index and Lemma 4.9 issues are correctness gaps in the written proofs, not instances of a result reducing by construction to its own input, so they do not raise the circularity score.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

The paper rests on standard CAT(0)/Hadamard facts rather than on fitted or invented quantities. No free parameters are fitted to data; the step-size sequences α_k and λ_n are explicit hypotheses, not fitted numbers.

assumptions (5)
  • standard math Complete CAT(0) spaces are uniquely geodesic and their distance satisfies the metric convexity condition (CC), Eq. (7).
    Used throughout Lemma 4.2, Lemma 4.3, and Theorem 4.11; cited to [44, Lemma 2.4].
  • domain assumption The proximal mapping Jλ in Hadamard spaces exists, is unique, 1-Lipschitz/firmly nonexpansive, and Fix(Jλ)=argmin f.
    Used in Section 5 to justify the Hyperbolic HalpernGD optimizer; cited to Bacák [17, Prop. 2.2.24, Lemma 4.2.2].
  • domain assumption Halpern iteration in complete CAT(0) spaces converges strongly to the metric projection of the anchor onto Fix(T) under conditions (C1), (C2), (C4).
    Used in Section 5 to assert convergence of the proposed hyperbolic Halpern scheme; cited to Saejung [46].
  • standard math The Sabach–Shtern recursion lemma (Lemma 4.9) holds as stated.
    Used directly in the proof of Theorem 4.11; cited to [57].
  • domain assumption Every nonexpansive map on a nonempty closed bounded convex subset of a complete CAT(0) space has a fixed point.
    Invoked in Section 2 to justify the setting; cited to Kirk [13, Theorem 18, p. 207].

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Pith. "Pith review of On the Rate of Asymptotic Regularity of Iterative Methods for Nonexpansive Mappings in CAT(0) Spaces and Hyperbolic Optimization." pith.science (2026). https://pith.science/paper/REAHWX7O

@misc{pith2026251025363,
  author       = {Pith},
  title        = {Pith review of: On the Rate of Asymptotic Regularity of Iterative Methods for Nonexpansive Mappings in CAT(0) Spaces and Hyperbolic Optimization},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/REAHWX7O}},
  note         = {Machine review of arXiv:2510.25363}
}
abstract

The Krasnosel'ski\u{\i}--Mann and Halpern iterations are classical schemes for approximating fixed points of nonexpansive mappings in Banach spaces, and have been widely studied in more general frameworks such as $CAT(\kappa)$ and, more generally, geodesic spaces. Convergence results and convergence rate estimates in these nonlinear settings are already well established. The contribution of this paper is twofold: first, we extend to complete $CAT(0)$ spaces proof techniques originally developed in the linear setting of Banach and Hilbert spaces, thereby recovering the same asymptotic regularity bounds; second, we introduce a Halpern--type optimizer for hyperbolic optimization as a nonlinear counterpart of the Euclidean HalpernSGD scheme.

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Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

  1. Krasnosel'skii-Mann iterations beyond asymptotics: a combinatorial analysis

    math.NA 2026-07 accept novelty 6.0 of 10

    Explicit non-asymptotic residual bounds for Krasnosel'skii–Mann iterations of contractions, with a smooth recovery of the nonexpansive rate as the contraction factor tends to 1.

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