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arxiv: 2606.18301 · v1 · pith:2DZXXYRDnew · submitted 2026-06-16 · 💻 cs.CG · math.PR

Denoising Distances in Metric Measure Spaces

Han Huang , Pakawut Jiradilok , Elchanan Mossel This is my paper

Pith reviewed 2026-06-26 22:09 UTC · model grok-4.3

classification 💻 cs.CG math.PR
keywords denoising distancesmetric measure spaceslower phi-regularitylocalized clustersstatistical-computational gapnoisy distancesclustering algorithm
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The pith

Localized clusters around sampled points allow denoising of distances to fixed accuracy in near-linear time under lower phi-regularity.

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

The paper establishes an algorithm for denoising pairwise distances drawn from noisy samples in general metric measure spaces that obey a lower phi-regularity condition. The method works by extracting large localized clusters around each sampled point and using those clusters to recover distances to any prescribed fixed accuracy, running in near-linear time when the sample is dense. A separate, inefficient procedure reaches substantially higher accuracy, which indicates a statistical-computational gap that does not appear in the Riemannian manifold setting. Readers care because the result shows how to perform reliable distance recovery without assuming the data lie on a smooth manifold.

Core claim

We give an algorithm that extracts large localized clusters around every sampled point and uses them to denoise distances to any fixed accuracy, with near-linear running time in the dense fixed-accuracy regime. We also show how to achieve much higher accuracy with a non-efficient algorithm. This suggests that unlike the Riemannian case, denoising to higher accuracy in more general metric spaces has a statistical-computational gap.

What carries the argument

Extraction of large localized clusters around each sampled point, made possible by the lower phi-regularity condition, which then supplies the denoised distance estimates.

If this is right

  • Distances between points can be recovered to any fixed accuracy with near-linear runtime when the sample is dense.
  • Substantially higher accuracy requires a computationally inefficient procedure, establishing a statistical-computational separation.
  • The same cluster-extraction technique applies to metric measure spaces beyond Riemannian manifolds.
  • Denoising succeeds once the regularity condition holds, without needing manifold structure or curvature bounds.

Where Pith is reading between the lines

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

  • The observed computational gap may appear in other geometric inference tasks posed on abstract metric spaces rather than manifolds.
  • The cluster extraction step could be repurposed for related problems such as local dimension estimation in the same spaces.
  • Empirical tests on synthetic metric spaces could delineate the precise boundary where the regularity condition begins to suffice for efficient denoising.

Load-bearing premise

The underlying metric measure space satisfies the lower phi-regularity condition that makes accurate cluster extraction possible.

What would settle it

A concrete metric measure space obeying lower phi-regularity on which the cluster-extraction procedure returns distance estimates whose error exceeds the target fixed accuracy.

Figures

Figures reproduced from arXiv: 2606.18301 by Elchanan Mossel, Han Huang, Pakawut Jiradilok.

Figure 1
Figure 1. Figure 1: Large-distance non-identifiability for a locally informative link. Two latent spaces may have identical local geometry but different separation between components. If the link function is flat beyond the local scale rp, cross-component observations cannot distinguish the gaps R1 and R2. without counting multiplicity is bounded by exp  C log2  3 rϕ(r)  1 sr 2  . The constants C, c, and the required lowe… view at source ↗
Figure 2
Figure 2. Figure 2: A folding obstruction at the volumetric scale, illustrated for ϕ(r) = r. Both panels show the same space: a circle with a duplicated cap of length r, equipped with the intrinsic metric. The only displayed change is the location of y: in panel (a), y lies in the red copy A−; in panel (b), it lies in the matched blue copy A+. Every point outside A− ∪ A+ has the same distance to these two matched locations. T… view at source ↗
read the original abstract

Recent work studied the problem of finding clusters and denoising pairwise distances from noisy distances of points sampled on a manifold. We study the same problems in more general metric measure spaces under \lowerphiregularity{}. We give an algorithm that extracts large localized clusters around every sampled point and uses them to denoise distances to any fixed accuracy, with near-linear running time in the dense fixed-accuracy regime. We also show how to achieve much higher accuracy with a non-efficient algorithm. This suggests that unlike the Riemannian case, denoising to higher accuracy in more general metric spaces has a statistical-computational gap.

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 / 2 minor

Summary. The paper extends prior work on clustering and distance denoising from noisy samples on manifolds to general metric measure spaces satisfying a lower phi-regularity condition. It presents an algorithm that extracts large localized clusters around each sampled point and uses them to denoise distances to any fixed accuracy, achieving near-linear running time in the dense fixed-accuracy regime. A separate non-efficient procedure is given for achieving much higher accuracy. The contrast is used to suggest that, unlike the Riemannian case, denoising to higher accuracy in more general metric spaces exhibits a statistical-computational gap.

Significance. If the claims hold under the stated regularity condition, the work provides a concrete algorithmic route to fixed-accuracy denoising in a broader class of spaces than manifolds, together with an explicit near-linear runtime guarantee in the dense regime. The suggestion of a statistical-computational gap, even if only heuristic, identifies a potential distinction worth further investigation in geometric data analysis.

minor comments (2)
  1. The abstract refers to "lower phi-regularity" without a self-contained definition or citation; the main text should state the precise condition (including the function phi) at the first use.
  2. The phrase "near-linear running time in the dense fixed-accuracy regime" should be accompanied by an explicit statement of the dependence on the accuracy parameter epsilon and the sampling density.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the summary, which accurately captures the paper's contributions on extending distance denoising to metric measure spaces under lower phi-regularity, the near-linear algorithm for fixed accuracy, and the non-efficient procedure for higher accuracy suggesting a statistical-computational gap. No major comments were provided in the report.

Circularity Check

0 steps flagged

No significant circularity in derivation chain

full rationale

The provided abstract and context describe an algorithmic result conditioned on lower phi-regularity of metric measure spaces, with cluster extraction used to achieve fixed-accuracy denoising in near-linear time and a separate non-efficient procedure for higher accuracy. No equations, fitted parameters, or self-citations are exhibited that reduce any claimed prediction or uniqueness result to the inputs by construction. The statistical-computational gap is presented only as a suggestion from contrasting procedures, not as a derived equality. The derivation chain is therefore self-contained against external benchmarks with no load-bearing self-referential steps.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claims rest on the lower phi-regularity assumption for the metric measure space; no free parameters or invented entities are mentioned in the abstract.

axioms (1)
  • domain assumption The metric measure space satisfies lower phi-regularity.
    Invoked to guarantee that localized clusters can be extracted and used for denoising to fixed accuracy.

pith-pipeline@v0.9.1-grok · 5624 in / 1171 out tokens · 22322 ms · 2026-06-26T22:09:49.031923+00:00 · methodology

discussion (0)

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