Denoising data using convex relaxations

Aalok Gangopadhyay; Charles Fefferman; Hariharan Narayanan; Jonathan Marty; Matti Lassas

arxiv: 2605.02327 · v2 · submitted 2026-05-04 · 📊 stat.ME · cs.LG

Denoising data using convex relaxations

Charles Fefferman , Aalok Gangopadhyay , Matti Lassas , Jonathan Marty , Hariharan Narayanan This is my paper

Pith reviewed 2026-05-08 19:11 UTC · model grok-4.3

classification 📊 stat.ME cs.LG

keywords denoisingconvex relaxationmanifoldfinite-sample boundsGaussian noiseprincipal component analysissupporting hyperplanescryo-electron microscopy

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The pith

A convex-relaxation estimator projects noisy manifold observations onto their convex hull after PCA reduction and achieves finite-sample error bounds under a lower-mass condition.

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

The authors develop a method to denoise data points sampled from a low-dimensional manifold embedded in high-dimensional space when each observation is corrupted by isotropic Gaussian noise. Their estimator first reduces dimension with principal component analysis, then projects each noisy point onto the convex hull of the projected manifold points. They build a statistical oracle that recovers the supporting hyperplanes of this hull by examining empirical tail probabilities of the Gaussian noise, and they prove that this oracle succeeds with high probability when the latent distribution places enough mass near every relevant direction. The resulting denoiser then inherits explicit error bounds derived from least-squares projection risk and entropy numbers for convex sets. The same assumptions are shown to hold for a standard cryo-electron microscopy forward model.

Core claim

We study the problem of denoising observations Y_i = X_i + Z_i, where the latent variables X_i are sampled from a low-dimensional manifold in R^n and the noise variables Z_i are isotropic Gaussian. We propose a convex-relaxation estimator that first reduces dimension by principal component analysis and then projects the observations onto the convex hull of the projected latent manifold. We construct a statistical oracle that estimates its supporting hyperplanes from empirical Gaussian tail probabilities of the noisy sample. Under a lower-mass condition on the latent distribution, we prove finite-sample guarantees for the oracle and derive error bounds for the resulting denoiser. The analysis

What carries the argument

The convex-relaxation estimator formed by PCA dimension reduction followed by projection onto the convex hull of the projected latent points, with supporting hyperplanes recovered by an oracle that uses empirical Gaussian tail probabilities.

If this is right

The denoiser error is bounded by a term that combines the projection risk under convex constraints with the entropy of the convex hull.
The oracle recovers each supporting hyperplane with high probability once the sample size exceeds a threshold determined by the lower-mass condition.
The same covering-number and Lipschitz estimates used for the cryo-EM model allow the framework to be instantiated for other group-equivariant imaging operators.
Finite-sample guarantees hold uniformly over all directions once the lower-mass condition is satisfied.

Where Pith is reading between the lines

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

The method could be combined with iterative refinement steps that update the convex hull estimate after an initial denoising pass.
Similar tail-probability oracles might be adapted to sub-Gaussian or bounded noise models by replacing Gaussian tail functions with appropriate concentration inequalities.
In practice the PCA step could be replaced by a robust dimension-reduction procedure when the noise variance is large relative to the manifold scale.
The entropy bounds on convex hulls suggest that the sample complexity scales with the intrinsic dimension rather than ambient dimension.

Load-bearing premise

The latent distribution must place sufficient mass in every direction so that empirical Gaussian tail probabilities can accurately recover the supporting hyperplanes of the convex hull.

What would settle it

Simulations in which the latent distribution is concentrated away from some supporting hyperplane directions yet the estimated denoiser still achieves the claimed error rate would show the lower-mass condition is not necessary.

Figures

Figures reproduced from arXiv: 2605.02327 by Aalok Gangopadhyay, Charles Fefferman, Hariharan Narayanan, Jonathan Marty, Matti Lassas.

**Figure 1.** Figure 1: Visual for the experiment with 20000 random samples from view at source ↗

**Figure 2.** Figure 2: A stylised visual for the experiment when view at source ↗

**Figure 3.** Figure 3: Visual for the hypocycloid experiment where the source point is at a view at source ↗

**Figure 4.** Figure 4: Visual for the hypocycloid experiment where the source point is near view at source ↗

**Figure 5.** Figure 5: Visual for the hypocycloid experiment where the source point in the view at source ↗

**Figure 6.** Figure 6: Estimating the distance of H from M by counting the number of samples on the side of H that does not contain M. and any preimage x ∗ ∈ Π −1 S {x} ∩ M0 we have Π−1 S B(x, ϵ) ⊇ B(x ∗ , ϵ), hence µM(B(x, ϵ)) = µ0(Π−1 S B(x, ϵ)) ≥ µ0(B(x ∗ , ϵ)) ≥ cM0 ωdϵ d by (1).) Let us denote by dist(H, x), the ℓ2 distance between x and the nearest point y, where y ∈ H. Let us denote by dist(H,M), the ℓ2 distance between t… view at source ↗

read the original abstract

We study the problem of denoising observations \(Y_i=X_i+Z_i\), where the latent variables \(X_i\) are sampled from a low-dimensional manifold in \(\mathbb{R}^n\) and the noise variables \(Z_i\) are isotropic Gaussian. We propose a convex-relaxation estimator that first reduces dimension by principal component analysis and then projects the observations onto the convex hull of the projected latent manifold. We construct a statistical oracle that estimates its supporting hyperplanes from empirical Gaussian tail probabilities of the noisy sample. Under a lower-mass condition on the latent distribution, we prove finite-sample guarantees for the oracle and derive error bounds for the resulting denoiser. The analysis combines risk bounds for least-squares projection under convex constraints with entropy bounds for convex hulls. We also verify the assumptions of the framework for a Cryo-Electron Microscopy observation model by establishing suitable covering number and Lipschitz estimates for the associated group action and imaging operators.

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.

Desk Editor's Note private letter to a colleague

The paper gives finite-sample guarantees for a PCA-plus-convex-hull denoiser on manifolds under a lower-mass condition, with a concrete check for the Cryo-EM model.

read the letter

The main takeaway is that the authors build a denoiser by first running PCA on the noisy points, then projecting onto the convex hull of the projected manifold, and they supply an oracle that estimates the supporting hyperplanes from empirical Gaussian tail probabilities. Under the lower-mass condition they prove finite-sample bounds for both the oracle and the final estimator by combining standard risk bounds for convex-constrained least squares with entropy bounds on convex hulls. They also carry out the covering-number and Lipschitz calculations needed to confirm the framework applies to the Cryo-EM observation model. That verification step is useful and keeps the work from being purely abstract. The estimator itself is new in this specific combination, and the argument does not rely on circular fitting or invented parameters. The lower-mass condition is the clearest limitation. It is invoked to make the tail-based hyperplane estimates reliable, yet the paper gives little indication of how restrictive it is for common manifold distributions or how the bounds degrade when it fails. That leaves the practical scope somewhat unclear. The rest of the math follows from established convex-geometry and statistical-estimation tools, so the internal logic holds up. This is for people working on manifold denoising or inverse problems in imaging. A reader who needs finite-sample theory for convex relaxations in this setting will get something concrete from the bounds and the Cryo-EM check. It deserves a serious referee because the claims are specific, the application verification is present, and the assumptions are stated plainly. I would send it out for review.

Referee Report

1 major / 2 minor

Summary. The manuscript studies denoising of observations Y_i = X_i + Z_i, where the latent X_i are sampled from a low-dimensional manifold in R^n and Z_i are isotropic Gaussian. The proposed convex-relaxation estimator first applies PCA for dimension reduction and then projects the observations onto the convex hull of the projected latent manifold. A statistical oracle is constructed that estimates supporting hyperplanes from empirical Gaussian tail probabilities of the noisy sample. Under a lower-mass condition on the latent distribution, finite-sample guarantees are proved for the oracle and error bounds are derived for the denoiser. The analysis combines risk bounds for least-squares projection under convex constraints with entropy bounds for convex hulls. The framework assumptions are verified for a Cryo-EM observation model via covering-number and Lipschitz estimates for the group action and imaging operators.

Significance. If the central derivations hold, the work provides a theoretically grounded approach to manifold denoising that yields explicit finite-sample oracle guarantees and denoiser error bounds without heavy parameter tuning. The combination of PCA, convex-hull projection, and Gaussian-tail hyperplane estimation is a clean use of convex geometry and standard concentration tools. Credit is due for the explicit verification of covering numbers and Lipschitz constants in the Cryo-EM model, which makes the framework falsifiable and reproducible in that setting. The result, if correct, would strengthen the literature on convex relaxations for high-dimensional estimation problems with manifold structure.

major comments (1)

[Abstract and the section deriving the oracle guarantees] The lower-mass condition on the latent distribution is invoked to guarantee accurate supporting-hyperplane estimation from empirical Gaussian tails and is therefore load-bearing for the finite-sample oracle guarantees stated in the abstract. The manuscript should supply a precise statement of the condition (including any dependence on dimension or manifold curvature) together with a brief argument or numerical check showing that it holds for the Cryo-EM model beyond the covering-number verification already given.

minor comments (2)

[Method description] The transition from the PCA step to the convex-hull projection should be accompanied by an explicit statement of the dimension-reduction error term so that readers can see how it interacts with the subsequent entropy bounds.
[Notation and preliminaries] Notation for the projected manifold, its convex hull, and the supporting hyperplanes should be introduced once and used consistently; currently the abstract and later sections appear to reuse symbols without a central definition table.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading, positive assessment, and constructive feedback on our manuscript. We address the single major comment below and have revised the manuscript accordingly to strengthen the presentation of the lower-mass condition.

read point-by-point responses

Referee: [Abstract and the section deriving the oracle guarantees] The lower-mass condition on the latent distribution is invoked to guarantee accurate supporting-hyperplane estimation from empirical Gaussian tails and is therefore load-bearing for the finite-sample oracle guarantees stated in the abstract. The manuscript should supply a precise statement of the condition (including any dependence on dimension or manifold curvature) together with a brief argument or numerical check showing that it holds for the Cryo-EM model beyond the covering-number verification already given.

Authors: We agree that a more explicit statement of the lower-mass condition is warranted, as it underpins the oracle guarantees. In the revised manuscript we have inserted a precise definition (now appearing immediately before the oracle analysis) that quantifies the required mass lower bound in terms of the ambient dimension n, the intrinsic dimension d, the noise variance, and an upper bound on the manifold curvature. The definition is stated as: for every supporting hyperplane direction, the latent measure assigns mass at least c·(noise scale)·(covering number factor) to a tubular neighborhood whose width depends on the curvature radius. We also added a short paragraph verifying the condition for the Cryo-EM model by combining the existing Lipschitz bounds on the imaging operator and group action with a standard volume argument on the manifold; this shows that the projected distribution satisfies the mass requirement with probability 1−O(1/n) under the same covering-number hypotheses already used for the entropy bounds. No changes to the main theorems or proofs were required. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation is self-contained

full rationale

The paper's central claims rest on constructing an oracle that estimates supporting hyperplanes from empirical Gaussian tail probabilities, then proving finite-sample guarantees under an explicitly stated lower-mass condition on the latent distribution. Error bounds for the denoiser follow from combining standard risk bounds for convex-constrained least squares with entropy bounds on convex hulls, after PCA dimension reduction. These steps invoke no fitted parameters renamed as predictions, no self-definitional equations, and no load-bearing self-citations; the Cryo-EM verification uses independent covering-number and Lipschitz estimates. The derivation chain is therefore independent of its own outputs.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on the lower-mass condition as a domain assumption needed for the oracle guarantees; standard mathematical properties of convex hulls, Gaussian distributions, and entropy numbers are used but not invented here.

axioms (1)

domain assumption lower-mass condition on the latent distribution
Invoked in the abstract to obtain finite-sample guarantees for the oracle estimating supporting hyperplanes.

pith-pipeline@v0.9.0 · 5466 in / 1293 out tokens · 30707 ms · 2026-05-08T19:11:12.753907+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

Unrelated to RS distinguishability floor (Foundation.AbsoluteFloorClosure) absolute_floor_iff_bare_distinguishability unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

μ₀(B(x, ϵ)) > c_{M_0} ω_d ϵ^d for all x ∈ M_0 — lower-mass / Ahlfors regularity condition on the latent measure.
No overlap with Cost.FunctionalEquation (J-cost uniqueness) or BranchSelection (bilinear branch forcing); the paper's cost is squared Euclidean, not the ratio-symmetric J(x) = ½(x+x⁻¹)−1. washburn_uniqueness_aczel unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

The analysis combines risk bounds for least-squares projection under convex constraints (Chatterjee 2014) with entropy bounds for convex hulls (Dudley's entropy integral).
Foundation.AlexanderDuality (forces D=3 from circle linking) — no contact: paper's D is a tunable algorithmic parameter, not a forced spatial dimension. alexander_duality_circle_linking unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

D := min(n, ⌈c_{M_0}^{-1} ω_d^{-1} ϵ_0^{-d}⌉) — the reduced PCA dimension, set by covering-number scaling and free to be any integer.

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

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Escaping the local minima via simulated annealing: Optimization of approximately convex functions

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Fitting a manifold to data in the presence of large noise

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work page arXiv 2023

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Fitting a manifold of large reach to noisy data.Journal of Topology and Analysis, 17(02):315–396, 2025

Charles Fefferman, Sergei Ivanov, Matti Lassas, and Hariharan Narayanan. Fitting a manifold of large reach to noisy data.Journal of Topology and Analysis, 17(02):315–396, 2025

work page 2025

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Testing the manifold hypothesis.Journal of the American Mathematical Society, 29(4):983–1049, 2016

Charles Fefferman, Sanjoy Mitter, and Hariharan Narayanan. Testing the manifold hypothesis.Journal of the American Mathematical Society, 29(4):983–1049, 2016

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Genovese, Marco Perone-Pacifico, Isabella Verdinelli, and Larry Wasserman

Christopher R. Genovese, Marco Perone-Pacifico, Isabella Verdinelli, and Larry Wasserman. Manifold estimation and singular deconvolution under hausdorff loss.The Annals of Statistics, 40(2):941–963, 2012

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Genovese, Marco Perone-Pacifico, Isabella Verdinelli, and Larry Wasserman

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On extreme points of regular convex sets

Mark Krein and David Milman. On extreme points of regular convex sets. Studia Mathematica, 9:133–138, 1940

work page 1940

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On the complexity of bandit and derivative-free stochastic convex optimization

Ohad Shamir. On the complexity of bandit and derivative-free stochastic convex optimization. In Shai Shalev-Shwartz and Ingo Steinwart, editors, Proceedings of the 26th Annual Conference on Learning Theory (COLT 2013), volume 30 ofJMLR Workshop and Conference Proceedings, pages 3–24. JMLR.org, 2013

work page 2013

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Information-theoretic lower bounds for convex optimization with erroneous oracles

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work page 2015

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Note on refined dudley integral covering number bound.Unpublished

Karthik Sridharan and Nathan Srebro. Note on refined dudley integral covering number bound.Unpublished

work page

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Cambridge university press, 2018

Roman Vershynin.High-dimensional probability: An introduction with ap- plications in data science, volume 47. Cambridge university press, 2018

work page 2018

[15] [15]

Manifold fitting

Zhigang Yao, Jiaji Su, Bingjie Li, and Shing-Tung Yau. Manifold fitting. arXiv preprint arXiv:2304.07680, 2023

work page arXiv 2023

[16] [16]

Manifold fitting with CycleGAN.Proceedings of the National Academy of Sciences, 121(5):e2311436121, 2024

Zhigang Yao, Jiaji Su, and Shing-Tung Yau. Manifold fitting with CycleGAN.Proceedings of the National Academy of Sciences, 121(5):e2311436121, 2024

work page 2024

[17] [17]

Manifold fitting under unbounded noise

Zhigang Yao and Yuqing Xia. Manifold fitting under unbounded noise. Journal of Machine Learning Research, 26(45):1–55, 2025. A Miscellaneous lemmas Lemma 6.LetMbe the manifold as mentioned earlier withreach(M)≥τ andvolume(M)≤V. Letω d be the volume of a d-dimensional unit Euclidean ball. Then NC(3ϵ,M, ℓ n 2 )≤N P (3ϵ/2,M, ℓ n 2 )≤ V ωdϵd ifϵ≤τ /4 NC(3ϵ,M,...

work page 2025