{"id":"4169ea8f-dc52-43c3-b281-3605e8f96e68","arxiv_id":"2510.25363","paper_version":3,"verdict":"CONDITIONAL","confidence":"HIGH","novelty_score":4.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"KM and Halpern-type iterations in CAT(0) spaces have the same O(1/√n) and O(1/n) asymptotic-regularity rates as in Banach spaces, and a proximal Halpern scheme is proposed for hyperbolic optimization.","lead":"This paper extends asymptotic-regularity rate proofs for Krasnosel'skii–Mann and Halpern-type fixed-point iterations from Hilbert and Banach spaces to CAT(0) spaces, and sketches a hyperbolic optimizer based on the Halpern iteration. A generalist might read it to see whether hyperbolic deep learning can inherit acceleration guarantees from classical fixed-point theory.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Theorem 4.11's proof invokes Lemma 4.9, which as stated is false (counterexample b_k≡1, c_k≡0, a_1=100); applying it with α_k=2/((1−β)k) would give O(k), not O(1/k), so the Halpern/viscosity rate is not proven as written.","rationale":"The reader correctly identified a summation-index mismatch in Proposition 4.5 that blocks Theorem 4.1 as written; that is a valid, load-bearing, and repairable gap. But the reader's statement that 'Theorem 4.11's proof is essentially sound' is not accurate: Lemma 4.9 is false as stated, and even its intended application to the proof of Theorem 4.11 does not yield the claimed O(1/k) bound — the stated conclusion would give O(k) when b_k=α_k. The theorem is likely true (O(1/k) Halpern rates in Hadamard spaces are known), and the proof gap is probably repairable by correcting Lemma 4.9 and re-checking constants, so the appropriate verdict remains CONDITIONAL. I keep the reader's verdict unchanged but flag the additional gap, hence 'partial' agreement with the reader's weakest-assumption choice.","tokens_in":18710,"tokens_out":24886,"duration_ms":193504,"concrete_test":"Check Lemma 4.9 against the source lemma in Sabach–Shtern [57]. Then evaluate the lemma as stated with b_k≡1, γ=1, c_k≡0, M=1, a_1=100: the hypothesis is satisfied but the conclusion a_k ≤ M J/b_k = 2 fails at k=1, so the lemma is false. Next, substitute b_k=α_k=2/((1−β)k), γ=1−β, M=2C_xbar into the corrected lemma (or solve the recurrence directly) and recompute the bound on d(x_k,x_{k−1}) in Theorem 4.11 for β∈{0,0.5,0.9}; if the result is not ≤ 2J C_xbar/((1−β)k), the displayed estimates and the derived residual bound need revision.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The reader's Proposition 4.5 summation-index mismatch is real and load-bearing for Theorem 4.1; changing the first-term sum from k=1..n to k=m+1..n would close the induction, so it is a repairable typo. However, Theorem 4.11 has a more direct obstruction. Lemma 4.9 is stated as: if a_{k+1} ≤ (1−γb_{k+1})a_k + (b_k−b_{k+1})c_k with c_k≤M and b_k nonincreasing, then a_k ≤ M J/b_k with J=⌈2/γ⌉. This is false: take b_k≡1, c_k≡0, γ=1, M=1, and a_1=100; the hypothesis holds but the conclusion forces a_1≤2. Moreover, even if one ignores that counterexample, the application in Theorem 4.11 sets b_k=α_k=2/((1−β)k), so the stated conclusion gives a_k ≤ M J (1−β)k/2, which is O(k), not O(1/k). The subsequent line 'Therefore d(x_k,x_{k−1}) ≤ 2J C/((1−β)k)' does not follow from the lemma as written; it appears to require a corrected lemma of the form a_k ≤ M J b_k (or a direct induction), and the constants still need checking. Thus the O(1/k) viscosity/Halpern rate is not proven as written. The gap is local and likely repairable, but it is load-bearing for Theorem 4.11.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","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.","tokens_in":19088,"tokens_out":3124,"duration_ms":33323,"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":[{"comment":"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.","section":"Proposition 4.5"},{"comment":"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.","section":"Lemma 4.9 and Theorem 4.11"},{"comment":"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.","section":"Theorem 4.11, second estimate"}],"minor_comments":[{"comment":"Reference [57] is a duplicate of [53] (Sabach–Shtern); please consolidate. The abstract says 'chapter' where 'paper' is intended.","section":"General"},{"comment":"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.","section":"Remark 4.7"},{"comment":"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.","section":"Section 5"}],"recommendation":"major_revision","confidential_remarks":"The paper has two genuine load-bearing gaps: the summation-index mismatch in Proposition 4.5 and the false/misapplied Lemma 4.9. Both appear locally repairable, and the rest of the transfer argument is coherent, so the paper should not be rejected outright. I recommend major revision: the authors should be asked to correct the summation limits in Proposition 4.5 and to replace Lemma 4.9 with a correct lemma (and proof) whose application to α_k=2/((1-β)k) yields the claimed O(1/k) bound. The self-citations in [1,2] are motivational and do not carry the proof load, so there is no novelty-disclosure concern."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Two things to know. First, the KM-rate theorem (4.1) looks right; the reviewer's worry about a summation-index mismatch in Proposition 4.5 is a false alarm. The first term is bounded by a sum over k=1..n, which only adds nonnegative terms relative to the original k=m+1..n, so the induction does close. Second, the O(1/k) viscosity/Halpern result (Theorem 4.11) is not proven as written. Lemma 4.9 is false as stated: take b_k=1, c_k=0, gamma=1, a_1=100; the recursion holds but the conclusion forces a_1 <= 2. Even ignoring that counterexample, the lemma gives a_k <= M J / b_k, and with b_k = 2/((1-beta)k) that is O(k), not O(1/k). The claimed bound d(x_k,x_{k-1}) <= 2J C/((1-beta)k) does not follow. The asymptotic-regularity bound inherits the problem. What is genuinely useful: the machinery transferring Cominetti's probabilistic weights to the metric setting is clean, and the CAT(0) KM bound appears to be new. The paper honestly credits [55] and [56] for prior O(1/sqrt(k)) and O(1/k) results in related settings. The hyperbolic optimizer is a natural construction, but it is entirely theoretical--no experiments, and the superiority over RSGD is explicitly an expectation. The soft spots, in proportion: the Lemma 4.9 issue is load-bearing for half the paper. It may be repairable--a corrected lemma of the form a_k <= M J b_k, or a direct induction, could give the intended O(1/k) rate--but the constants need re-checking. The optimizer section is speculative by the paper's own admission, which is acceptable for a theory paper but should be framed as such. The citations are accurate and not inflated; the self-citations are motivational only. Who should read it: people working on fixed-point algorithms in CAT(0)/Hadamard spaces, and anyone designing hyperbolic optimizers. It deserves a serious referee because the questions are important and the KM part may be correct; but the referee should focus on fixing Lemma 4.9 and tracing its consequences. I'd send it to review, with a clear expectation of major revision.","headline":"The KM-rate theorem in CAT(0) looks correct, but the viscosity/Halpern O(1/k) proof rests on a false lemma as stated.","tokens_in":722,"tokens_out":784,"would_cite":false,"duration_ms":38801,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["47H09","47H10"],"pacs":[],"model":"deepseek-v4-flash","headline":"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","keywords":["CAT(0) spaces","Krasnosel'skii–Mann iteration","Halpern iteration","nonexpansive mappings","asymptotic regularity","convergence rate","Hadamard spaces","hyperbolic optimization"],"falsifier":"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.","tokens_in":18542,"feed_emoji":"📐","tokens_out":12641,"duration_ms":116356,"temperature":0.7,"pith_summary":"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.","feed_headline":"CAT(0) spaces keep Euclidean optimizer speeds","feed_subtitle":"Halpern iteration stays O(1/k); Krasnoselskii–Mann stays O(1/√n) in curved spaces.","key_machinery":"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.","core_discovery":"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","pith_inferences":["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."],"forward_implications":["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."],"fun_headline_variants":["CAT(0) spaces keep classical regularity rates","Euclidean optimizer speeds carry to curved spaces","Same O(1/k) and O(1/√n) bounds in CAT(0)","Hyperbolic optimizer matches Euclidean convergence"],"cache_read_input_tokens":2304,"weakest_assumption_plain":"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.","fun_headline_variants_meta":{"raw":{"variants":["CAT(0) spaces keep classical regularity rates","Euclidean optimizer speeds carry to curved spaces","Same O(1/k) and O(1/√n) bounds in CAT(0)","Hyperbolic optimizer matches Euclidean convergence"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000226,"raw_usage":{"total_tokens":1297,"prompt_tokens":726,"completion_tokens":571,"prompt_tokens_details":{"cached_tokens":256},"prompt_cache_hit_tokens":256,"prompt_cache_miss_tokens":470,"completion_tokens_details":{"reasoning_tokens":505}},"tokens_in":470,"tokens_out":571,"duration_ms":6362,"temperature":1.0,"reasoning_tokens":505,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-04T07:31:08.744896+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"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.","supporting_citations":[],"review_version":1}