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One-Bit Clustering for Two Component Sub-Gaussian Mixture Models

T0 review · 2 major / 2 minor · reviewed 2026-06-26 · grok-4.3

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Pith's one-line read A variant of Lloyd's algorithm on one-bit dithered quantized samples clusters two-component sub-Gaussian mixtures with misclassification rates decaying exponentially in signal-to-noise ratio at rates comparable to full data.

desk verdict This paper gives the first one-bit clustering method for two-component sub-Gaussian mixtures, with misclassification rates decaying exponentially at rates close to the unquantized case under a non-spikiness condition on centers, plus exact recovery above a separation threshold only a log factor worse. read the letter →

arxiv 2606.21873 v1 pith:C54OOOYW submitted 2026-06-20 cs.IT eess.SPmath.ITstat.ML

Junren Chen , Yun Yang This is my paper · ORCID
classification cs.ITeess.SPmath.ITstat.ML
keywords one-bitclusteringsub-GaussianmixturemodelsLloyd'salgorithmditheredquantizationmisclassificationrateexactrecoveryminimaxlowerbound
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

The paper establishes that one-bit clustering is possible for two-component sub-Gaussian mixture models by quantizing each sample entry to a single bit using a dithered quantizer. A variant of Lloyd's algorithm then achieves misclassification rates that decay exponentially in the signal-to-noise ratio at rates comparable to those with full-precision data, provided the cluster centers satisfy a mild non-spikiness condition. This guarantee further yields exact cluster recovery whenever the separation between centers exceeds the information-theoretic threshold for unquantized data by only a logarithmic factor. When the dimension is high, a preliminary random rotation renders the non-spikiness condition likely to hold, and matching lower bounds show the constants are sharp.

What carries the argument

Variant of Lloyd's algorithm applied to one-bit dithered quantized samples

What would settle it

Fix the signal-to-noise ratio and increase the spikiness of the centers while measuring whether the misclassification rate stops decaying exponentially.

Watch

Extended reading notes

Core claim

Under a mild non-spikiness condition on the cluster centers, a variant of Lloyd's algorithm applied after one-bit dithered quantization achieves a misclassification rate that decays exponentially with a signal-to-noise ratio comparable to the unquantized setting. This implies exact recovery under an explicit separation condition exceeding the optimal unquantized threshold by only a logarithmic factor. When the dimension p is sufficiently large, the non-spikiness condition can be enforced by a random rotation using a Haar distributed matrix prior to quantization, holding with high probability for p ≳ 1 in partial recovery and p ≳ log n log log n in exact recovery. A minimax lower bound establ

Load-bearing premise

The cluster centers must satisfy a mild non-spikiness condition; if they are spiky the exponential decay and exact recovery may fail.

Editorial extensions

If this is right

  • Misclassification rate decays exponentially in the signal-to-noise ratio at a rate comparable to the unquantized case.
  • Exact recovery holds when separation exceeds the unquantized optimum by only a logarithmic factor.
  • Random rotation before quantization enforces non-spikiness with high probability when p is large enough.
  • The achieved rates match the minimax lower bound up to constants.

Reading between the lines

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

  • Dithering is essential to keep the one-bit observations informative about the cluster means.
  • The same quantization-plus-Lloyd pipeline may apply to other mixture estimation tasks once the non-spikiness issue is handled.
  • Storage or communication cost in distributed clustering can drop by roughly the original bit depth per coordinate.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

2 major / 2 minor

Summary. The manuscript proposes the first one-bit clustering method for two-component sub-Gaussian mixture models, using dithered one-bit quantization per entry. Under a mild non-spikiness condition on the cluster centers, a variant of Lloyd's algorithm is shown to achieve a misclassification rate decaying exponentially in the signal-to-noise ratio at a rate comparable to the unquantized setting. This implies exact recovery under an explicit separation condition exceeding the optimal unquantized threshold by only a logarithmic factor. When dimension p is large, a random rotation via a Haar-distributed matrix enforces the non-spikiness condition with high probability (p ≳ 1 for partial recovery; p ≳ log n log log n for exact recovery). A matching minimax lower bound establishes sharpness of the misclassification rate and separation condition, supported by numerical experiments.

Significance. If the derivations hold, the work establishes the first rigorous performance guarantees for one-bit quantized clustering in this model class, achieving near-optimal exponential rates and exact recovery thresholds. Notable strengths include the matching lower bound with sharp constants and the explicit random-rotation argument that quantifies the dimension threshold needed to enforce non-spikiness. These elements provide a concrete bridge between quantized sensing and classical mixture-model theory.

major comments (2)
  1. [Abstract and rotation argument section] The non-spikiness condition is load-bearing for both the Lloyd-variant upper bound and the exact-recovery claim (as stated in the abstract). The random-rotation argument is invoked to restore the condition, but the manuscript must explicitly derive the p ≳ log n log log n threshold for exact recovery and confirm that the failure probability is controlled uniformly over the two centers; without this derivation the conditional nature of the central claim remains incompletely substantiated.
  2. [Exact recovery theorem] The separation condition for exact recovery is claimed to exceed the unquantized optimum by only a logarithmic factor. The relevant theorem should contain an explicit side-by-side comparison of the two thresholds (including the precise logarithmic term) so that the “only logarithmic” statement can be verified directly against the lower bound.
minor comments (2)
  1. [Preliminaries] Notation for the dithered quantizer and the precise definition of the non-spikiness parameter should be introduced once in a dedicated preliminary section rather than scattered across the abstract and main text.
  2. [Numerical results] The numerical section would benefit from an explicit statement of the parameter regimes (n, p, SNR) used in the simulations relative to the theoretical thresholds.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the positive assessment and constructive suggestions. We address the two major comments below and will revise the manuscript accordingly to improve clarity and completeness.

read point-by-point responses

  1. Referee: [Abstract and rotation argument section] The non-spikiness condition is load-bearing for both the Lloyd-variant upper bound and the exact-recovery claim (as stated in the abstract). The random-rotation argument is invoked to restore the condition, but the manuscript must explicitly derive the p ≳ log n log log n threshold for exact recovery and confirm that the failure probability is controlled uniformly over the two centers; without this derivation the conditional nature of the central claim remains incompletely substantiated.

    Authors: We agree that an explicit derivation of the dimension threshold strengthens the presentation. In the revised version we will expand the random-rotation argument (currently sketched in the relevant section) to include a self-contained proof that the non-spikiness condition holds with high probability for p ≳ log n log log n under exact recovery, with the failure probability controlled uniformly over both centers via a union bound. The updated text will also clarify how this probability is absorbed into the overall success probability of the algorithm. revision: yes

  2. Referee: [Exact recovery theorem] The separation condition for exact recovery is claimed to exceed the unquantized optimum by only a logarithmic factor. The relevant theorem should contain an explicit side-by-side comparison of the two thresholds (including the precise logarithmic term) so that the “only logarithmic” statement can be verified directly against the lower bound.

    Authors: We will revise the exact-recovery theorem statement and its surrounding discussion to include a direct side-by-side display of the one-bit separation threshold and the corresponding unquantized threshold (both taken from the matching lower bound), explicitly isolating the logarithmic factor. This will allow immediate verification of the claimed gap without cross-referencing other results. revision: yes

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity; theoretical guarantees derived from explicit assumptions and analysis

full rationale

The paper states performance guarantees under a mild non-spikiness condition on cluster centers and derives exponential misclassification decay plus exact recovery via analysis of a Lloyd variant and random rotation (for p sufficiently large). No quoted steps reduce by construction to fitted inputs, self-definitions, or load-bearing self-citations; the non-spikiness premise is an external modeling assumption, not derived from the target rates, and the rotation argument is shown to hold w.h.p. under dimension thresholds. The minimax lower bound further indicates independent content. This is the standard case of a self-contained theoretical derivation.

Assumptions & free parameters 0 free parameters · 2 assumptions · 0 invented entities

The central claims rest on the two-component sub-Gaussian mixture model and the non-spikiness condition on centers; these are domain assumptions that enable the analysis but are not derived within the paper.

assumptions (2)
  • domain assumption Data follows a two-component sub-Gaussian mixture model
    This is the model class stated in the title and abstract as the setting for the clustering problem.
  • domain assumption Mild non-spikiness condition on the cluster centers
    Invoked as necessary for the misclassification guarantees, exact recovery, and the random rotation argument to hold with high probability.

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Cite this review

Pith. "Pith review of One-Bit Clustering for Two Component Sub-Gaussian Mixture Models." pith.science (2026). https://pith.science/paper/C54OOOYW

@misc{pith2026260621873,
  author       = {Pith},
  title        = {Pith review of: One-Bit Clustering for Two Component Sub-Gaussian Mixture Models},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/C54OOOYW}},
  note         = {Machine review of arXiv:2606.21873}
}
abstract

Clustering is a fundamental problem in statistics and machine learning. We propose the first one-bit clustering method for two-component sub-Gaussian mixture models. The method uses only one bit per entry of each sample obtained via a dithered quantizer. Under a mild non-spikiness condition on the cluster centers, we show that a variant of Lloyd's algorithm achieves a misclassification rate that decays exponentially with a signal-to-noise ratio comparable to that in the unquantized setting. This result further implies exact recovery under an explicit separation condition, which exceeds the optimal threshold for unquantized data by only a logarithmic factor. When the dimension $p$ is sufficiently large, the non-spikiness condition can be enforced by applying a random rotation using a Haar distributed matrix prior to quantization. In particular, it holds with high probability when $p \gtrsim 1$ for partial recovery and $p \gtrsim \log n \log\log n$ for exact recovery, where $n$ is the sample size. We also establish a minimax lower bound, showing that the misclassification rate and separation condition exhibit sharp constants in general. Numerical results are provided to corroborate the theory and demonstrate the efficacy of the proposed method.

Figures

Figures reproduced from arXiv: 2606.21873 by the authors.

Figure 1
Figure 1. Phase transitions and comparison with classical clustering. [PITH_FULL_IMAGE:figures/full_fig_p010_1.png] view at source ↗
Figure 2
Figure 2. Graphical illustration of Equation (A.1): the red curve Q2(a) and the blue curve sign(a) are identical when |a| < 2. We now proceed the analysis with the quantized samples X˜ i = Q2λ(Xi + τi), i = 1, · · · , n, which can be arranged in the matrix Y˜ = [X˜ 1, · · · , X˜ n]. We then proceed to analyze the following two-stage procedure, replacing Y˙ in Algorithm 1 by Y˜ : 13 [PITH_FULL_IMAGE:figures/full_fig_p013_2.png] view at source ↗

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Reference graph

Works this paper leans on

37 extracted references · 2 canonical work pages

  1. [1]

    An lp theory of pca and spectral clustering.The Annals of Statistics, 50(4):2359–2385, 2022

    Emmanuel Abbe, Jianqing Fan, and Kaizheng Wang. An lp theory of pca and spectral clustering.The Annals of Statistics, 50(4):2359–2385, 2022

    2022

  2. [2]

    Robust mean estimation under quantization.arXiv preprint arXiv:2601.07074, 2026

    Pedro Abdalla and Junren Chen. Robust mean estimation under quantization.arXiv preprint arXiv:2601.07074, 2026

    work page arXiv 2026

  3. [3]

    Courier Corporation, 2012

    Robert B Ash.Information theory. Courier Corporation, 2012

    2012

  4. [4]

    Distributed adaptive gaussian mean estimation with unknown variance: Interactive protocol helps adaptation.The Annals of Statistics, 50(4):1992–2020, 2022

    T Tony Cai and Hongji Wei. Distributed adaptive gaussian mean estimation with unknown variance: Interactive protocol helps adaptation.The Annals of Statistics, 50(4):1992–2020, 2022

    1992

  5. [5]

    A parameter-free two-bit covariance estimator with improved operator norm error rate.Applied and Computational Harmonic Analysis, page 101774, 2025

    Junren Chen and Michael K Ng. A parameter-free two-bit covariance estimator with improved operator norm error rate.Applied and Computational Harmonic Analysis, page 101774, 2025

    2025

  6. [6]

    Ng, and Di Wang

    Junren Chen, Cheng-Long Wang, Michael K. Ng, and Di Wang. High dimensional sta- tistical estimation under uniformly dithered one-bit quantization.IEEE Transactions on Information Theory, 69(8):5151–5187, 2023

    2023

  7. [7]

    One-bit phase retrieval: Optimal rates and efficient algo- rithms.IEEE Transactions on Information Theory, 2026

    Junren Chen and Ming Yuan. One-bit phase retrieval: Optimal rates and efficient algo- rithms.IEEE Transactions on Information Theory, 2026

    2026

  8. [8]

    Cutoff for exact recovery of gaussian mixture models.IEEE Transactions on Information Theory, 67(6):4223–4238, 2021

    Xiaohui Chen and Yun Yang. Cutoff for exact recovery of gaussian mixture models.IEEE Transactions on Information Theory, 67(6):4223–4238, 2021

    2021

Show all 37 references
  1. [9]

    Nonconvex optimization meets low-rank matrix factorization: An overview.IEEE Transactions on Signal Processing, 67(20):5239–5269, 2019

    Yuejie Chi, Yue M Lu, and Yuxin Chen. Nonconvex optimization meets low-rank matrix factorization: An overview.IEEE Transactions on Signal Processing, 67(20):5239–5269, 2019

    2019

  2. [10]

    1-bit matrix completion.Information and Inference: A Journal of the IMA, 3(3):189–223, 2014

    Mark A Davenport, Yaniv Plan, Ewout Van Den Berg, and Mary Wootters. 1-bit matrix completion.Information and Inference: A Journal of the IMA, 3(3):189–223, 2014

    2014

  3. [11]

    An overview of low-rank matrix recovery from incomplete observations.IEEE Journal of Selected Topics in Signal Processing, 10(4):608– 622, 2016

    Mark A Davenport and Justin Romberg. An overview of low-rank matrix recovery from incomplete observations.IEEE Journal of Selected Topics in Signal Processing, 10(4):608– 622, 2016

    2016

  4. [12]

    Covariance estimation under one-bit quantization.The Annals of Statistics, 50(6):3538–3562, 2022

    Sjoerd Dirksen, Johannes Maly, and Holger Rauhut. Covariance estimation under one-bit quantization.The Annals of Statistics, 50(6):3538–3562, 2022

    2022

  5. [13]

    Non-gaussian hyperplane tessellations and robust one-bit compressed sensing.Journal of the European Mathematical Society, 23(9):2913– 2947, 2021

    Sjoerd Dirksen and Shahar Mendelson. Non-gaussian hyperplane tessellations and robust one-bit compressed sensing.Journal of the European Mathematical Society, 23(9):2913– 2947, 2021

    2021

  6. [14]

    Phase retrieval by binary questions: Which complementary subspace is closer?Constructive Approximation, 56(1):1–33, 2022

    Dylan Domel-White and Bernhard G Bodmann. Phase retrieval by binary questions: Which complementary subspace is closer?Constructive Approximation, 56(1):1–33, 2022

    2022

  7. [15]

    Hidden integrality of sdp relaxations for sub-gaussian mix- ture models

    Yingjie Fei and Yudong Chen. Hidden integrality of sdp relaxations for sub-gaussian mix- ture models. InConference On Learning Theory, pages 1931–1965. PMLR, 2018

    1931

  8. [16]

    Zhang, and Harrison H

    Chao Gao, Zongming Ma, Anderson Y. Zhang, and Harrison H. Zhou. Community detec- tion in degree-corrected block models.Annals of Statistics, 46(5):2153–2185, 2018

    2018

  9. [17]

    Partial recovery bounds for clustering with the relaxedk-means.Mathematical Statistics and Learning, 1(3):317–374, 2019

    Christophe Giraud and Nicolas Verzelen. Partial recovery bounds for clustering with the relaxedk-means.Mathematical Statistics and Learning, 1(3):317–374, 2019. 11

    2019

  10. [18]

    Dithered quantizers.IEEE Transactions on Information Theory, 39(3):805–812, 1993

    Robert M Gray and Thomas G Stockham. Dithered quantizers.IEEE Transactions on Information Theory, 39(3):805–812, 1993

    1993

  11. [19]

    Robust 1- bit compressive sensing via binary stable embeddings of sparse vectors.IEEE Transactions on Information Theory, 59(4):2082–2102, 2013

    Laurent Jacques, Jason N Laska, Petros T Boufounos, and Richard G Baraniuk. Robust 1- bit compressive sensing via binary stable embeddings of sparse vectors.IEEE Transactions on Information Theory, 59(4):2082–2102, 2013

  12. [20]

    Mean estimation from one-bit measurements.IEEE Transactions on Information Theory, 68(9):6276–6296, 2022

    Alon Kipnis and John C Duchi. Mean estimation from one-bit measurements.IEEE Transactions on Information Theory, 68(9):6276–6296, 2022

    2022

  13. [21]

    Optimality of spectral clus- tering in the gaussian mixture model.The Annals of Statistics, 49(5):2506–2530, 2021

    Matthias L¨ offler, Anderson Y Zhang, and Harrison H Zhou. Optimality of spectral clus- tering in the gaussian mixture model.The Annals of Statistics, 49(5):2506–2530, 2021

    2021

  14. [22]

    Statistical and computational guarantees of lloyd’s algorithm and its variants.arXiv preprint arXiv:1612.02099, 2016

    Yu Lu and Harrison H Zhou. Statistical and computational guarantees of lloyd’s algorithm and its variants.arXiv preprint arXiv:1612.02099, 2016

    2016 arXiv

  15. [23]

    Sharp optimal recovery in the two component gaussian mixture model

    Mohamed Ndaoud. Sharp optimal recovery in the two component gaussian mixture model. The Annals of Statistics, 50(4):2096–2126, 2022

  16. [24]

    Restricted strong convexity and weighted matrix completion: Optimal bounds with noise.Journal of Machine Learning Research, 13(1):1665–1697, 2012

    Sahand Negahban and Martin J Wainwright. Restricted strong convexity and weighted matrix completion: Optimal bounds with noise.Journal of Machine Learning Research, 13(1):1665–1697, 2012

    2012

  17. [25]

    Approximating k-means-type clustering via semidefinite pro- gramming.SIAM journal on optimization, 18(1):186–205, 2007

    Jiming Peng and Yu Wei. Approximating k-means-type clustering via semidefinite pro- gramming.SIAM journal on optimization, 18(1):186–205, 2007

    2007

  18. [26]

    Robust 1-bit compressed sensing and sparse logistic regression: A convex programming approach.IEEE Transactions on Information Theory, 59(1):482–494, 2012

    Yaniv Plan and Roman Vershynin. Robust 1-bit compressed sensing and sparse logistic regression: A convex programming approach.IEEE Transactions on Information Theory, 59(1):482–494, 2012

    2012

  19. [27]

    Picture coding using pseudo-random noise.IRE Transactions on In- formation Theory, 8(2):145–154, 1962

    Lawrence Roberts. Picture coding using pseudo-random noise.IRE Transactions on In- formation Theory, 8(2):145–154, 1962

    1962

  20. [28]

    Sketching for distributed deep learning: A sharper analysis.Advances in Neural Infor- mation Processing Systems, 37:6417–6447, 2024

    Mayank Shrivastava, Berivan Isik, Qiaobo Li, Sanmi Koyejo, and Arindam Banerjee. Sketching for distributed deep learning: A sharper analysis.Advances in Neural Infor- mation Processing Systems, 37:6417–6447, 2024

    2024

  21. [29]

    The generalized lasso for sub-gaussian measurements with dithered quantization.IEEE Transactions on Information Theory, 66(4):2487–2500, 2020

    Christos Thrampoulidis and Ankit Singh Rawat. The generalized lasso for sub-gaussian measurements with dithered quantization.IEEE Transactions on Information Theory, 66(4):2487–2500, 2020

    2020

  22. [30]

    Cambridge University Press, 2018

    Roman Vershynin.High-dimensional probability: An introduction with applications in data science, volume 47. Cambridge University Press, 2018

    2018

  23. [31]

    Randomly initialized em algorithm for two-component gaussian mixture achieves near optimality ino( √n) iterations.Mathematical Statistics & Learning, 4, 2021

    Yihong Wu and Harrison H Zhou. Randomly initialized em algorithm for two-component gaussian mixture achieves near optimality ino( √n) iterations.Mathematical Statistics & Learning, 4, 2021

    2021

  24. [32]

    Quantized compressive sensing with rip matrices: The benefit of dithering.Information and Inference: A Journal of the IMA, 9(3):543–586, 2020

    Chunlei Xu and Laurent Jacques. Quantized compressive sensing with rip matrices: The benefit of dithering.Information and Inference: A Journal of the IMA, 9(3):543–586, 2020. 12 A Proof of Theorem 2.1 (Partial recovery) Proof.Asλ ˙Xi is the surrogate ofX i, we define the overa...

    2020

  25. [33]

    Let ˜vbe the leading eigenvector ofH( ˜YT ˜Y) and let ˜η0 = sign(˜v)

  26. [34]

    We lete i = ˜Xi−Xi fori∈[n], then we have thatei is zero-mean andO(λ) sub-Gaussian (cf

    Run ˜ηk+1 = sign(H( ˜YT ˜Y)˜ηk) to obtain the sequence{˜ηi}i≥0. We lete i = ˜Xi−Xi fori∈[n], then we have thatei is zero-mean andO(λ) sub-Gaussian (cf. Lemma F.12). We further letu i =e i +εi, then the observations ˜Xi can be expressed as ˜Xi =X i +ei =ηiθ+εi +ei =X i +ui, or ...

  27. [35]

    Recall that δ1 d=Q 2λ(θ+ε1 +τ1)−θ, 22 we shall introduce a surrogate ofδ1 as ˜δ1 :=λsign(θ+ε1 +τ1)−E [ λsign(θ+ε1 +τ1) ] = (˜δ11,···,˜δ1p)T

    +P(Γ∩ˆB∩ˆB1) ≤1 n2 + 2 exp(−10r2 n,λ) +P ( ⟨δ1, ˆξ⟩<− ( 1−O ( r−1 n,λ+ (logn n )1/4 )) ∥θ∥2 2√ ∥θ∥2 2 + pλ2 n )    :=P∗∗ (A.45) ▶by Equations (A.40), (A.42) and (A.44) To establish sharp bound onP ∗∗, we need a reduction to the one-bit case. Recall that δ1 d=Q 2λ(θ+ε1 +τ1)...

  28. [36]

    = sign( ˙YT j 1) attains the minimum of the right-hand side of (E.1). Then, it is not hard to see that EπE|¯ηj( ˙Xj)−ηj|= 2P ( Binomial(p,P−1)> p 2 ) ≥1√2p exp ( −pDKL (1 2∥P−1 )) (E.2) 28 where the last inequality is due to the anti-concentration bound for binomial variable (...

  29. [37]

    Also, byλ≥(3 logp p )1/2+σ √ 2(1 +ν) log(np)≥ ∥θR∥∞+σ √ 2(1 +ν) log(np), Lemma F.2 along with a union bound gives P( ˆB2)≥P ( ∥Rε1∥∞≤σ √ 2(1 +ν) log(np) ) ≥1−1 n1+νpν. Therefore, for someϵn,λ=O(r−1 n,λ+ (logn n )1/4) + 1 log1/4(np), we have P∗∗=P ( ⟨˜δ′ 1, ˆξ⟩ λ ≤−(1−ϵn,λ)rn,λ...

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