{"id":"0b7c482f-3e62-40b4-925b-30c4357a1a02","arxiv_id":"2506.18761","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":2,"one_line_summary":"Two rounds of local averaging move a noisy high-dimensional sample to within O(σ√d) of a d-dimensional manifold, proven for the first time at noise levels comparable to the manifold's reach.","lead":"This paper proves that a simple and widely used procedure, averaging nearby noisy data points, can move them close to the hidden low-dimensional structure even when the noise is large. The result gives the first theoretical guarantee for this behavior at noise levels comparable to the manifold's reach, supporting local averaging as a preprocessing step.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Stage II's extra condition 1/κ ≥ C5σ√(D log D) is load-bearing: at the advertised σ√D ≈ τ regime the proof's own condition (A.12) fails, so the refined bound is not established there.","rationale":"The reader's weakest_assumption identifies exactly the condition I find load-bearing: the Stage II assumption 1/κ ≥ C5 σ√(D log D) is stronger than the abstract's advertised σ√D ≈ τ regime. My analysis sharpens this from a disclosure problem to a proof-level obstruction: at the advertised boundary, the proof's own condition (A.12) / Lemma C.1 condition (C.5) is violated, causing the second-round signal-average error to degrade by √log D. This does not invalidate the theorem as stated, because the theorem explicitly assumes the stronger condition; it does mean the headline claim in the abstract and the 'first analysis at σ√D ≈ τ' framing overstate what Stage II proves. Since the reader already rendered a CONDITIONAL verdict and the required repair is to either add the stronger noise condition to the abstract/framing or extend the proof to handle the boundary, no change to the verdict is needed. The paper still provides a legitimate first analysis of local averaging in a regime that is far more permissive than prior exponential-small-noise assumptions, but the advertised boundary regime is not achieved for the refined two-stage bound.","tokens_in":44415,"tokens_out":21717,"duration_ms":214972,"concrete_test":"Analytically instantiate the advertised boundary regime: set 1/κ = τ = C σ√D (so σ√D ≈ τ) and use the Stage II choices R2² = σ²(D−3+D^{3/4}+2C D^{5/12}) with d(q1,M)² bounded as in (A.84). Compute L = log(diam(M)/š⋆,∥) + dκdiam(M) + d log(1/(κs⋆,∥)) and check whether Lemma C.1 condition (C.5) holds: š⋆,∥² + σ²√D √L ≤ c4 š⋆,∥². The left side exceeds the right by a factor ≳ D^{1/12}√log D for large D, showing the proof cannot be invoked in the advertised regime. A complementary numerical check: simulate the two-round algorithm on a low-dimensional manifold (e.g., S¹ or S² in D ≈ 1000) at σ√D = τ/2 versus σ√D = τ/(2√log D) and measure d(q2,M); the former should exhibit the √log D degradation in the absence of the extra assumption.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central theorem's Stage II requires 1/κ ≥ C5 σ√(D log D). Since the reach satisfies τ ≤ 1/κ, this confines the refined bound to σ√D ≲ τ/√log D, a factor √log D below the advertised 'noise comparable to reach' regime. This is not cosmetic. In the Stage II parameter range, d(q1,M)² ≈ σ²D^{3/4} and R2² − σ²(D−3) ≈ σ²D^{3/4}, so s⋆,∥² ≈ σ²D^{5/12} and š⋆,∥² ≈ σ²D^{5/12}. Lemma C.1 / condition A.12 requires š² + √(log(diam/š) + dκdiam + d log(1/(κs⋆,∥))) σ²√D ≤ c4 š². At the advertised boundary 1/κ ≈ τ ≈ σ√D, the square-root term is at least c σ²√(D log D), because diam/š ≥ D^{7/24}; its ratio to š² ≈ σ²D^{5/12} is ≳ D^{1/12}√log D → ∞. Hence the proof's verification of Theorem B.1/C.1 fails exactly in the advertised regime. Proceeding without the extra assumption, the signal-average bound (C.6) contributes κσ²√D√L ≈ σ√(d log D), exceeding the claimed σ√d bound by √log D. Thus the abstract's claim of a first analysis at σ√D ≈ τ is not established for Stage II; the theorem only supports the refined bound under the stronger hypothesis 1/κ ≥ C5 σ√(D log D).","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper analyzes a two-stage minibatch local averaging scheme for estimating a d-dimensional submanifold M ⊂ R^D from samples x_i = x_i,♮ + z_i with z_i ~ N(0, σ²I_D). In Stage I, averaging a minibatch of points in a ball of radius R1 around a noisy initialization is claimed to give d(q1, M) ≤ Cσ√(d(κdiam(M) + log D)) under σ√D ≤ c1τ, where τ is the reach. In Stage II, a Gaussian perturbation ϑ ~ N(0, σ²D^{-1/4}I) is injected, a second minibatch in a ball of radius R2 is averaged, and the refined bound d(q2, M) ≤ Cσ√(d(1 + κdiam(M)/log D)) is claimed under the additional hypothesis 1/κ ≥ C5σ√(D log D). The technical core is a subgaussian analysis of the grouping probability h(s), the probability that a noisy point falls in an extrinsic ball, obtained via incomplete-gamma-function bounds near its phase transition; this yields sharp bounds on the conditional noise expectation and the conditional intrinsic squared distance of grouped points, combined with Bernstein and vector-Hoeffding inequalities. The paper claims this is the first analysis of local averaging accuracy in the regime σ√D ≈ τ.","tokens_in":44766,"tokens_out":53902,"duration_ms":456658,"significance":"If fully established, the refined bound would be a valuable contribution to manifold estimation: the algorithm is simple, the radii and minibatch sizes are explicit functions of σ, d, D, κ and diam(M) rather than fitted quantities, and the phase-transition estimates for h(s) (Lemmas B.2–B.7) appear to be new, carefully proved, and likely reusable for tangent-space and curvature estimation. The paper is honest about its limitations (Section 6), including the lack of geometric-quantity estimation and the breakdown of the subgaussian tool when the landmark is very close to the manifold. The Stage I bound under σ√D ≤ c1τ is a solid contribution in its own right even if the Stage II regime claim must be narrowed. The main liabilities are the mismatch between the advertised and the proved Stage II regime, an unverified scaling condition in the Stage II proof of (A.9), and the reliance on two same-group preprints for load-bearing T_max bounds. These issues are fixable within the manuscript's scope, but they directly affect the paper's central novelty claim, so the present version is not ready.","major_comments":[{"comment":"The abstract's headline claim is not supported by the hypotheses of Theorem 4.1. Stage II requires the additional hypothesis 1/κ ≥ C5σ√(D log D), and the reach satisfies τ ≤ 1/κ (a relation the paper itself uses, e.g., in Lemma C.2). When the reach is curvature-limited (τ ≈ 1/κ, as for a round sphere factor), this condition confines the refined bound to σ√D ≲ τ/√(log D), a factor √(log D) below the advertised 'noise comparable to the reach' regime; the abstract states the refined d(q̂, M) bound immediately after claiming the first analysis in the regime σ√D ≈ τ, conflating the Stage I regime (where σ√D ≤ c1τ is genuinely established) with the Stage II regime. I also could not locate any step in the Stage (c) proof that invokes 1/κ ≥ C5σ√(D log D): the verifications of (A.8)–(A.12) and the bounds (A.57)–(A.108) use only σ√D ≤ c1τ, σD^{20} ≥ 1/κ, and D^{1/12} ≥ C1 max{κ̄²d, κdiam(M)d}. If the condition is unnecessary, it should be deleted or weakened to the requirement 1/κ ≥ CσD^{5/24} forced by (A.10), which would restore the advertised regime and make the abstract accurate; if it is necessary, the exact step using it must be identified. I note that the stress-test concern that condition (A.12) fails at the advertised boundary does not land: the RHS of (A.12) is c4š²⋆, of order σ²D^{3/4} in Stage II, so the term σ²√(D log D) is negligible (ratio √(log D)/D^{1/4} → 0) rather than divergent; the regime gap comes from the stated hypothesis and from (A.9), not from (A.12).","section":"Abstract; §4, Theorem 4.1, Stage II"},{"comment":"The verification of the second-round conditions of Theorems B.1 and C.1 is incomplete, and the step verifying (A.9) has a genuine scaling gap. In Stage II, s⋆² = σ²(D^{3/4} + 2C D^{5/12}), so (A.9) requires C2(log D + κddiam(M) + d log(1/(κs⋆,∥)))σ²D^{2/3} ≤ s⋆², i.e., the bracket must be O(D^{1/12}). The text bounds the third bracket term only by C d log D (via σD^{20} ≥ 1/κ and s⋆,∥ ≥ CσD^{5/24}), while the hypothesis D^{1/12} ≥ C1κ̄²d, κ̄ ≥ 1, gives only d ≤ D^{1/12}/C1; these do not imply d log D ≤ cD^{1/12}. Along the maximal allowed choice d ≈ D^{1/12}/(C1κ̄²), the bracket is Θ(d log D) = Θ(D^{1/12} log D), exceeding the required size by the factor Θ(log D). This is not merely a constants issue, since the discrepancy is in the scaling with D. The subsequent sentence 'Similar to round 1' cannot discharge (A.10)–(A.12) either, because their right-hand sides in the second round scale as σ²D^{3/4} rather than σ²D as in the first round. The fix is to add an explicit large-D assumption such as D^{1/12} ≥ C8(d log D + log D + κdiam(M)d) and to spell out the verification of (A.10)–(A.12) for the second round.","section":"Appendix A, Stage (c), verification of (A.9)"},{"comment":"The refined bound is not self-contained at a load-bearing point. The lower bound on d(q1, M) in (A.70)–(A.84), which sets the scale of the second-round radius, uses the bound Tmax,ϑ ≤ Cκ̄σD^{-1/2}√d from Lemma A.4 of [10] and the statement 'which along with Lemma B.5 from [10]' at (A.67)–(A.68); Lemma D.2 similarly uses Theorem 3 of [31] for E[Tmax] ≤ Cκ̄σ√d. References [10] (arXiv:2502.16819) and [31] (TPoPT) are same-group preprints whose proofs are not reproduced in this manuscript. Since the Stage II claim is the paper's central contribution, the authors should either prove these short statements (which appear to follow from the ε-net/covering arguments already present in Section B) or cite refereed versions; otherwise the 'first analysis' claim rests on unpublished work.","section":"Appendix A, Eqs. (A.64)–(A.69), and Lemma D.2"}],"minor_comments":[{"comment":"There are several typos: 'decribe' in §5.1, 'denoiing' in the acknowledgments, 'signficantly' in §4 after Eq. (4.2), and 'optimziation' in the title of [31].","section":"Throughout"},{"comment":"The injected perturbation ϑ ∼ N(0, σ²D^{-1/4}I) is introduced without comment in Algorithm 1; the reader only learns in footnote 6 that it is a purely technical device. Since the paper motivates the method by practical simplicity, the main text should flag this at the algorithm.","section":"Algorithm 1 / §3"},{"comment":"The caption says that d(q2, M) is bounded by σ√d 'regardless of ambient dimension D'; the theorem's bound is Cσ√(d(1 + κdiam(M)/log D)), which is independent of D only in scaling, so the caption should be reworded.","section":"Figure 1 caption"},{"comment":"The relation τ ≤ 1/κ between the reach and the curvature bound is used implicitly (Lemma C.2) but never stated; it should be stated and cited near the problem setup, especially since the Stage II regime discussion depends on it.","section":"Lemma C.2 / problem setup"},{"comment":"The proof of Stirling's formula via the de Moivre–Laplace theorem spans several pages; a standard reference for Stirling's bound would shorten the appendix considerably.","section":"Appendix D, Lemma D.5"},{"comment":"Reference [31] is cited as 'preprint, 2023' without an arXiv identifier; please provide one for verifiability.","section":"References"},{"comment":"These sketched bounds appear to refer to the second round, but the round is not specified; the round-1 noise-average bound (A.29) lacks the D^{-1/8} factor that appears in (5.3), so an explicit statement would prevent confusion.","section":"§5.1, Eqs. (5.2)–(5.3)"}],"recommendation":"major_revision","confidential_remarks":"The two load-bearing external dependencies ([10], [31]) are same-group preprints; I would ask the authors to prove the relevant T_max statements in an appendix or confirm refereed versions exist. The abstract overstates the Stage II regime; after resolving the (A.9) gap and either deleting or justifying the hypothesis 1/κ ≥ C5σ√(D log D), the paper would be a solid contribution to manifold estimation and denoising. The paper's fit with the journal is good."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The short version: the paper has a genuinely new technical tool and a credible first-stage analysis, but the headline second-stage claim is not established as written, and the abstract oversells the regime.\n\nWhat is actually new: the subgaussian bounds for the grouping probability h(s) — a chi-square CDF in the ambient dimension — are the right kind of tool for this problem, and they do give a new Stage I coarse-localization result under σ√D ≲ τ. That regime is excluded by [5,6] and much broader than [7]. The two-round scheme is simple, the constants are explicit, and there is no fitted circularity in the main distance bound.\n\nWhere it softens. First, Stage II in Theorem 4.1 assumes 1/κ ≥ C5 σ√(D log D). Since τ ≤ 1/κ, the refined bound is actually confined to σ√D ≲ τ/√log D, not σ√D ≈ τ. That by itself would be a somewhat stronger-regime problem, but then the proof's own condition A.12 does not close. With R2 set as in Stage II, the relevant quantities are š² ≈ σ²D^{5/12} and the square-root term in A.12 ≈ σ²√(D d log D) (up to constants and the κ diam term). For large D the ratio is D^{1/12}√(d log D), so A.12 cannot hold as D grows unless d is absurdly small — it isn't. The proof does not actually verify A.12 for Stage II; it says \"similar to round 1\" and moves on. If my arithmetic is right, that is a gap in the proof of Theorem 4.1, not just a mismatch with the abstract.\n\nTwo smaller issues. The Stage II proof leans on Lemma A.4 of [10] and Theorem 3 of [31], both same-group preprints, with no proof here. Plausible, but borrowed. And the paper never counts the total stream length needed to fill the minibatches, which matters for a preprocessing claim.\n\nWho this is for: anyone working on provable manifold denoising and fitting. The subgaussian phase-transition analysis is worth reading on its own, and Stage I may stand after repair. But the paper currently overclaims the first analysis at σ√D ≈ τ.\n\nRecommendation: send to peer review, but with a clear expectation of major revision. The authors should either prove Stage II under a hypothesis that actually satisfies A.12, or restate the theorem honestly with the stronger noise gap, and make the borrowed T_max bounds self-contained or at least visible.","headline":"Genuinely useful subgaussian phase-transition tool and a credible Stage I analysis, but the advertised Stage II regime is not proven and the paper overstates it.","tokens_in":45387,"tokens_out":9956,"would_cite":true,"duration_ms":98660,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["62R30","60E15","62G05"],"pacs":[],"model":"deepseek-v4-flash","headline":"This paper proves that two rounds of local averaging bring noisy points to within $O(\\sigma\\sqrt{d})$ of a low-dimensional manifold, even when the noise is comparable to the manifold's reach.","keywords":["local averaging","manifold estimation","Gaussian noise","reach","high-noise regime","subgaussian concentration","phase transition","landmarking"],"falsifier":"Run the two-round algorithm on a low-curvature manifold with noise chosen so that $\\sigma\\sqrt{D}\\approx\\tau$ but $\\sigma\\sqrt{D\\log D}>1/\\kappa$, using the theorem's stated radii and minibatch sizes, and measure $d(q_2,\\mathcal{M})$ over many trials; if the distance exceeds $C\\sigma\\sqrt{d(1+\\kappa\\,\\mathrm{diam}(\\mathcal{M})/\\log D)}$ with non-negligible probability, the theorem's advertised regime would be falsified.","tokens_in":44165,"feed_emoji":"📐","tokens_out":10005,"duration_ms":90088,"temperature":0.7,"pith_summary":"This paper proves a finite-sample guarantee for a two-round, minibatch local-averaging algorithm when noisy samples lie near a $d$-dimensional manifold in $\\mathbb{R}^D$. It is the first analysis of local averaging in the regime where the typical noise magnitude $\\sigma\\sqrt{D}$ is comparable to the manifold's reach $\\tau$. With appropriately chosen acceptance radii and minibatch sizes, one round brings a landmark to distance $O(\\sigma\\sqrt{d(\\log D+\\kappa\\,\\mathrm{diam}(\\mathcal{M}))})$ from the manifold, and a second round improves this to $O(\\sigma\\sqrt{d(1+\\kappa\\,\\mathrm{diam}(\\mathcal{M})/\\log D)})$, removing the dependence on ambient dimension up to a log factor. The result matters because local averaging is the core of many provable manifold-fitting and denoising methods, which previously had guarantees only under much smaller noise.","feed_headline":"Two local averages put noisy points within σ√d of the manifold","feed_subtitle":"A simple two-round averaging method provably works when noise is comparable to the manifold's reach.","key_machinery":"The load-bearing object is the grouping probability $h(s)=P(g_2^2+\\cdots+g_D^2\\le (R^2-s^2)/\\sigma^2)$ for i.i.d. standard Gaussians $g_i$, which is the probability that a clean point at distance $s$ from the current landmark is accepted into the ball. Its sharp phase transition at $s_\\star^2=R^2-\\sigma^2(D-3)$ captures the geometry of acceptance: in high ambient dimension the noise shell makes points at that distance the main contributors. The paper derives two-sided subgaussian bounds for $h$ and its derivative near $s_\\star$, convolves them with the one-dimensional Gaussian along the landmark direction to bound the conditional mean of accepted noise, and combines this with a curvature lemma that bounds the signal average's distance to the manifold by $\\kappa$ times the average squared intrinsic distance. An injected Gaussian perturbation after round one keeps the landmark at distance at least $\\sigma D^{3/8}$ from the manifold, so the second-round analysis stays in the normal-injectivity neighborhood where projection derivatives are controlled.","core_discovery":"The paper's central claim is Theorem 4.1: under lower bounds on the ambient dimension $D$, the curvature bound $\\kappa$, and conditions $\\sigma D\\ge1$, $\\sigma D^{20}\\ge1/\\kappa$, $\\sigma\\sqrt{D}\\le c_1\\tau$, the two-stage procedure in Algorithm 1 outputs a landmark $q_2$ with $d(q_2,\\mathcal{M})\\le C\\sigma\\sqrt{d(1+\\kappa\\,\\mathrm{diam}(\\mathcal{M})/\\log D)}$ with probability at least $1-9e^{-c_2 d}$. The proof splits the average into a clean-signal average and a conditioned-noise average, controls the signal term by a curvature lemma, and controls the noise term through new subgaussian bounds on the probability $h(s)$ that a noisy point is accepted into the ball $B(q,R)$. The function $h(s)$ has a sharp phase transition at $s_\\star^2=R^2-\\sigma^2(D-3)$, and the paper shows the convolution of $h$ with the Gaussian noise density is subgaussian around that transition, which is what makes the high-noise regime tractable.","pith_inferences":["The advertised regime is narrower than the abstract suggests: the Stage II condition $1/\\kappa\\ge C_5\\sigma\\sqrt{D\\log D}$ is stronger than $\\sigma\\sqrt{D}\\le c_1\\tau$ because the reach obeys $\\tau\\le1/\\kappa$, so at fixed geometry the second-round bound requires noise about a factor $\\sqrt{\\log D}$ smaller.","The phase-transition analysis of $h(s)$ is not tied to the two-round scheme; the same subgaussian bounds should transfer to mean shift, nonlocal-means patch averaging, and graph-Laplacian weight construction under Gaussian noise.","The injected perturbation is present only to keep the landmark away from the manifold; replacing it with explicit regularization could yield a proof for arbitrarily many averaging rounds.","The paper's own secant-bundle heuristic suggests the true limit may be $E[\\Sigma]\\approx\\tau$ rather than $\\sigma\\sqrt{D}\\approx\\tau$; a direct test is whether trimmed or robust local averages can be analyzed at that higher noise boundary."],"forward_implications":["Local averaging can serve as a preprocessing step that extends low-noise manifold-fitting guarantees to noise levels up to $\\sigma\\sqrt{D}\\lesssim\\tau$.","Pairwise distances between clean signals can be estimated from local averages with error $O(\\sigma d^{1/4}D^{1/4}(\\log D)^{1/4})$, improving on the raw noisy-point distance in high dimensions.","Replacing noisy samples by their local averages turns an $O(\\sigma\\sqrt{D})$ net for the manifold into an $O(\\sigma(dD\\log D)^{1/4})$ net, giving a discretization for downstream tasks such as denoising new samples.","Because the algorithm only needs minibatches of accepted samples, it admits an online implementation and scales better with the ambient dimension than prior provable large-noise manifold-fitting algorithms."],"supporting_citations":[{"why":"Low-noise manifold-fitting method whose validity the averaged landmarks can extend; it requires exponentially small noise, so it sets the baseline the paper relaxes.","marker":"[5]"},{"why":"Manifold fitting under unbounded noise; another provable estimator whose small-noise assumption the two-round averaging preprocessor is meant to relax.","marker":"[6]"},{"why":"Local geometric structure estimation from noisy data; its noise regime $\\sigma\\sqrt{D}\\lesssim\\tau/\\sqrt{D\\log D}$ is the weaker regime the paper improves on.","marker":"[7]"},{"why":"Prior algorithm for fitting manifolds of large reach in the large-noise regime; its complexity motivates the simpler averaging analysis.","marker":"[23]"},{"why":"Local regularization of noisy point clouds; provides the pairwise-distance estimation result that Corollary 4.2 compares against and improves.","marker":"[9]"},{"why":"Fast manifold denoising by tunneling optimization; supplies the tangent-max and projection lemmas used in the second round, as well as a net-building application.","marker":"[10]"},{"why":"Regularity of projection onto differentiable manifolds; used to control how the injected perturbation moves the projection during round two.","marker":"[29]"},{"why":"High-dimensional probability text from which the proof takes the Bernstein concentration inequality for the signal-average finite-sample bound.","marker":"[27]"},{"why":"Tail bound for chi-square random variables used to capture the upper tail of the grouping probability $h(s)$.","marker":"[30]"},{"why":"Matrix Hoeffding inequality used to obtain the vector Hoeffding bound for the averaged noise vectors.","marker":"[32]"}],"fun_headline_variants":["First proof: local averaging works under noise as large as reach","Local averaging provably accurate when noise matches manifold's reach","Two-round averaging provably distills manifolds in high-noise regimes","Theory: local averaging accurate even with noise near manifold reach"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The second refinement stage is proved only when $1/\\kappa\\ge C_5\\sigma\\sqrt{D\\log D}$, a condition stricter than the headline $\\sigma\\sqrt{D}\\le c_1\\tau$ because the reach satisfies $\\tau\\le1/\\kappa$; if the noise lies between those two levels, the claimed $\\sigma\\sqrt{d}$ second-round accuracy is not established.","fun_headline_variants_meta":{"raw":{"variants":["First proof: local averaging works under noise as large as reach","Local averaging provably accurate when noise matches manifold's reach","Two-round averaging provably distills manifolds in high-noise regimes","Theory: local averaging accurate even with noise near manifold reach"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000381,"raw_usage":{"total_tokens":2109,"prompt_tokens":1118,"completion_tokens":991,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":734,"completion_tokens_details":{"reasoning_tokens":921}},"tokens_in":734,"tokens_out":991,"duration_ms":11023,"temperature":1.0,"reasoning_tokens":921,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-15T18:44:10.033432+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Run the two-round algorithm on a low-curvature manifold with noise chosen so that $\\sigma\\sqrt{D}\\approx\\tau$ but $\\sigma\\sqrt{D\\log D}>1/\\kappa$, using the theorem's stated radii and minibatch sizes, and measure $d(q_2,\\mathcal{M})$ over many trials; if the distance exceeds $C\\sigma\\sqrt{d(1+\\kappa\\,\\mathrm{diam}(\\mathcal{M})/\\log D)}$ with non-negligible probability, the theorem's advertised regime would be falsified.","supporting_citations":[{"cited_title":"Manifold fitting, 2023","cited_arxiv_id":null,"evidence_quote":"Low-noise manifold-fitting method whose validity the averaged landmarks can extend; it requires exponentially small noise, so it sets the baseline the paper relaxes."},{"cited_title":"Manifold fitting under unbounded noise.Journal of Machine Learning Research, 26(45):1–55, 2025","cited_arxiv_id":null,"evidence_quote":"Manifold fitting under unbounded noise; another provable estimator whose small-noise assumption the two-round averaging preprocessor is meant to relax."},{"cited_title":"Estimation of local geometric structure on manifolds from noisy data.Journal of Machine Learning Research, 26(64):1–89, 2025","cited_arxiv_id":null,"evidence_quote":"Local geometric structure estimation from noisy data; its noise regime $\\sigma\\sqrt{D}\\lesssim\\tau/\\sqrt{D\\log D}$ is the weaker regime the paper improves on."},{"cited_title":"Local regularization of noisy point clouds: Improved global geometric estimates and data analysis.Journal of Machine Learning Research, 20(136):1–37, 2019","cited_arxiv_id":null,"evidence_quote":"Local regularization of noisy point clouds; provides the pairwise-distance estimation result that Corollary 4.2 compares against and improves."},{"cited_title":"Fast, Accurate Manifold Denoising by Tunneling Riemannian Optimization","cited_arxiv_id":"2502.16819","evidence_quote":"Fast manifold denoising by tunneling optimization; supplies the tangent-max and projection lemmas used in the second round, as well as a net-building application."},{"cited_title":"Existence, uniqueness and regularity of the projection onto differentiable manifolds.Annals of global analysis and geometry, 60(3):559–587, 2021","cited_arxiv_id":null,"evidence_quote":"Regularity of projection onto differentiable manifolds; used to control how the injected perturbation moves the projection during round two."},{"cited_title":"1 Nmb NmbX ℓ=1 zℓ−E","cited_arxiv_id":null,"evidence_quote":"Matrix Hoeffding inequality used to obtain the vector Hoeffding bound for the averaged noise vectors."}],"review_version":2}