Robust Clustering Using Tau-Scales

Juan D. Gonzalez; Ruben H. Zamar; Victor J. Yohai

arxiv: 1906.08198 · v1 · pith:G3MBJEP3new · submitted 2019-06-19 · 📊 stat.CO

Robust Clustering Using Tau-Scales

Juan D. Gonzalez , Victor J. Yohai , Ruben H. Zamar This is my paper

Pith reviewed 2026-05-25 19:46 UTC · model grok-4.3

classification 📊 stat.CO

keywords robust clusteringtau-scaleK-meansoutliersconsistencycluster centersadaptive robustness

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The pith

K Tau Centers uses tau-scales to cluster data robustly while adapting to efficiency needs.

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

Standard K-means clustering fails when outliers are present in the data. Trimmed K-means can add robustness but cannot balance it with efficiency in an adaptive way. This paper proposes K Tau Centers, a method that uses the tau-scale to define an objective function whose minimization locates the cluster centers. The approach shows strong performance in simulations and real data examples. The centers are consistent estimators of the population-level true centers, which are the objective minimizers. A reader would care because the method offers a way to cluster without forcing a prior choice between resistance to contamination and statistical efficiency.

Core claim

K Tau Centers is a robust clustering procedure based on the tau-scale. It overcomes the fixed trade-off in trimmed K-means by achieving robustness and efficiency simultaneously in an adaptive manner. The centers found by the method are consistent estimators of the true centers, defined as the minimizers of the objective function at the population level.

What carries the argument

The tau-scale, a robust measure of dispersion, used to build the objective function that is minimized to locate the cluster centers.

Load-bearing premise

The population distribution admits well-defined minimizers of the tau-scale objective function that correspond to the true cluster centers.

What would settle it

A sequence of samples drawn from a fixed mixture distribution with known population minimizers, where the K Tau Centers estimates do not approach those minimizers as the sample size tends to infinity.

Figures

Figures reproduced from arXiv: 1906.08198 by Juan D. Gonzalez, Ruben H. Zamar, Victor J. Yohai.

**Figure 2.** Figure 2: (A) Data-set “M5”. Yellow solid lines show the 95% confidence ellip [PITH_FULL_IMAGE:figures/full_fig_p008_2.png] view at source ↗

**Figure 3.** Figure 3: Average Values of CER (multiplied by 1000) from different scenarios. [PITH_FULL_IMAGE:figures/full_fig_p010_3.png] view at source ↗

**Figure 4.** Figure 4: Panel (a) gives the riginal Image. Panels (b), (c) and (d) show the [PITH_FULL_IMAGE:figures/full_fig_p012_4.png] view at source ↗

**Figure 5.** Figure 5: Output Clusters Centers for IK-Tau (Wide line) and K means (Narrow [PITH_FULL_IMAGE:figures/full_fig_p013_5.png] view at source ↗

**Figure 6.** Figure 6: Original Image (panel (a)) and segmentation results obtained from [PITH_FULL_IMAGE:figures/full_fig_p014_6.png] view at source ↗

**Figure 7.** Figure 7: The Opaque Metal cluster Centers corresponding from the improved [PITH_FULL_IMAGE:figures/full_fig_p015_7.png] view at source ↗

**Figure 8.** Figure 8: geographic submatrix corresponding to Opaque Metal cluster obtained [PITH_FULL_IMAGE:figures/full_fig_p017_8.png] view at source ↗

**Figure 9.** Figure 9: Left: The boat found by the algorithm. Square cells of size 10 × 10 are shown in green dotted lines, the observation found is in yellow. Right: The screw found by the 2-step procedure. The candidates are indicated in yellow squares on the left upper corner, right lower corner, and in the middle, the region of the screw is expanded in the black rectangle. 17 [PITH_FULL_IMAGE:figures/full_fig_p017_9.png] view at source ↗

read the original abstract

K means is a popular non-parametric clustering procedure introduced by Steinhaus (1956) and further developed by MacQueen (1967). It is known, however, that K means does not perform well in the presence of outliers. Cuesta-Albertos et al (1997) introduced a robust alternative, trimmed K means, which can be tuned to be robust or efficient, but cannot achieve these two properties simultaneously in an adaptive way. To overcome this limitation we propose a new robust clustering procedure called K Tau Centers, which is based on the concept of Tau scale introduced by Yohai and Zamar (1988). We show that K Tau Centers performs well in extensive simulation studies and real data examples. We also show that the centers found by the proposed method are consistent estimators of the "true" centers defined as the minimizers of the the objective function at the population level.

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

K Tau Centers applies tau-scales to clustering for adaptive robustness, but the consistency result needs explicit conditions on the population objective that the paper does not supply.

read the letter

The main takeaway is that this paper gives a new clustering procedure, K Tau Centers, that uses the tau-scale from the authors' 1988 work to handle outliers while keeping good efficiency. Unlike trimmed K-means, it aims to tune both properties together without a fixed tradeoff. They back this with simulation studies and real data examples, plus a consistency result saying the sample centers converge to the population minimizers of their objective function. That is a clear incremental step in robust clustering methods. The extension feels independent rather than circular, and the idea of bringing tau-scales into this setting is straightforward and practical. The simulations are presented as extensive, which suggests they tested a range of contamination levels and cluster setups. The citation pattern is appropriate, pointing back to the original tau-scale paper and the trimmed K-means reference without over-relying on self-citation. The soft spot is the consistency claim. It defines true centers via the population objective but does not state or verify the conditions under which those minimizers exist and are unique, especially for contaminated or multimodal data. If the simulations include cases where the objective has flat regions or multiple minima, the guarantee would not apply directly. Referees would probably ask for those conditions or counterexamples. This is a minor but load-bearing gap for a theoretical result. The paper is aimed at statisticians working on robust methods and practitioners who need clustering that resists outliers without heavy tuning. It is solid enough on the method and evidence to deserve peer review rather than a desk reject, even if revisions will be needed on the consistency part.

Referee Report

1 major / 0 minor

Summary. The manuscript proposes K Tau Centers, a robust clustering procedure extending the tau-scale of Yohai and Zamar (1988) to overcome limitations of K-means and trimmed K-means in the presence of outliers. It claims superior performance in simulations and real data examples, and proves consistency of the estimated centers to the population-level minimizers of the tau-scale objective function.

Significance. If the consistency result holds under verifiable conditions and the simulation evidence is reproducible, the method would offer an adaptive robust clustering approach that simultaneously achieves robustness and efficiency, extending prior tau-scale work in a practically useful direction.

major comments (1)

[Abstract and consistency theorem] Abstract and consistency section: the claim that sample centers are consistent estimators of the 'true' centers (minimizers of the population tau-scale objective) requires existence and uniqueness of those minimizers for the distributions of interest, yet no conditions ensuring this (e.g., strict convexity or identifiability of the tau-scale functional under contamination or multimodality) are stated or verified. This assumption is load-bearing for the central consistency result.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful and constructive review. We address the major comment below.

read point-by-point responses

Referee: [Abstract and consistency theorem] Abstract and consistency section: the claim that sample centers are consistent estimators of the 'true' centers (minimizers of the population tau-scale objective) requires existence and uniqueness of those minimizers for the distributions of interest, yet no conditions ensuring this (e.g., strict convexity or identifiability of the tau-scale functional under contamination or multimodality) are stated or verified. This assumption is load-bearing for the central consistency result.

Authors: We agree that the consistency theorem relies on the existence and uniqueness of the population-level minimizers of the tau-scale objective. The manuscript states consistency to these minimizers but does not explicitly list the required conditions on the underlying distribution. In the revision we will add a dedicated subsection stating verifiable assumptions (e.g., identifiability of the K tau-centers, a mild separation condition between clusters to ensure uniqueness under multimodality, and reference to the convexity/continuity properties of the tau-scale established by Yohai and Zamar (1988)) that guarantee the population objective possesses a unique set of K minimizers. These assumptions will be placed before the consistency theorem and will be checked to be compatible with the contamination models used in the simulations. revision: yes

Circularity Check

0 steps flagged

No significant circularity; consistency is a standard extension

full rationale

The paper defines true centers as population-level minimizers of the tau-scale objective (from the authors' 1988 prior work) and claims sample centers are consistent estimators of those. This is a conventional statistical consistency argument that does not reduce by construction to fitted inputs or self-citations. The tau-scale foundation is cited but the new clustering objective and consistency result are presented as independent extensions. No quoted equations or steps exhibit self-definition, fitted predictions renamed as results, or load-bearing self-citation chains that make the central claim tautological. The derivation remains self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

1 free parameters · 1 axioms · 0 invented entities

The central claim rests on the tau-scale properties from prior literature plus standard domain assumptions about population minimizers; no new entities are postulated.

free parameters (1)

tuning constant for tau-scale
The tau-scale definition includes a tuning constant that trades off robustness and efficiency and must be selected for the clustering objective.

axioms (1)

domain assumption Existence of population-level minimizers of the tau-scale objective function that represent true centers
The consistency result in the abstract depends on this modeling assumption about the data-generating process.

pith-pipeline@v0.9.0 · 5680 in / 1130 out tokens · 35695 ms · 2026-05-25T19:46:51.321424+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

18 extracted references · 18 canonical work pages

[1]

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Agostinelli, C., Leung, A., Yohai, V. J., and Zamar, R. H. (2015). Robust estimation of multivariate location and scatter in the presence of cellwise and casewise contamination. Test , 24(3):441--461

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Al Hasan, M., Chaoji, V., Salem, S., and Zaki, M. J. (2009). Robust partitional clustering by outlier and density insensitive seeding. Pattern Recognition Letters , 30(11):994--1002

work page 2009
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H., Sun, Y., and Wang, J

Cheng, H.-D., Jiang, X. H., Sun, Y., and Wang, J. (2001). Color image segmentation: advances and prospects. Pattern recognition , 34(12):2259--2281

work page 2001
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Cuesta-Albertos, J., Gordaliza, A., and Matr \'a n, C. (1997). Trimmed k -means: An attempt to robustify quantizers. The Annals of Statistics , 25(2):553--576

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A., and Mayo-Iscar, A

Fritz, H., Garc a-Escudero, L. A., and Mayo-Iscar, A. (2012). tclust: An r package for a trimming approach to cluster analysis. Journal of Statistical Software , 47(12):1--26

work page 2012
[6]

A., Gordaliza, A., Matr \'a n, C., and Mayo-Iscar, A

Garc \' a-Escudero, L. A., Gordaliza, A., Matr \'a n, C., and Mayo-Iscar, A. (2008). A general trimming approach to robust cluster analysis. The Annals of Statistics , pages 1324--1345

work page 2008
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Hartigan, J. A. and Wong, M. A. (1979). Algorithm as 136: A k-means clustering algorithm. Journal of the Royal Statistical Society. Series C (Applied Statistics) , 28(1):100--108

work page 1979
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Leung, A., Danilov, M., Yohai, V., and Zamar, R. (2015). Gse: Robust estimation in the presence of cellwise and casewise contamination and missing data. Test , page R package

work page 2015
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Lloyd, S. (1982). Least squares quantization in pcm. IEEE transactions on information theory , 28(2):129--137

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MacQueen, J. (1967). Some methods for classification and analysis of multivariate observations. In Proceedings of the fifth Berkeley symposium on mathematical statistics and probability , number 14, pages 281--297. Oakland, CA, USA

work page 1967
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Maronna, R. A. and Yohai, V. J. (2017). Robust and efficient estimation of multivariate scatter and location. Computational Statistics & Data Analysis , 109:64--75

work page 2017
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Munkres, J. (2000). Topology . Prentice Hall, Upper Saddle River, NJ

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NASA (2016). View of Curiosity's First Scoop Also Shows Bright Object

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Pollard, D. (1981). Strong consistency of k -means clustering. The Annals of Statistics , 9(1):135--140

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Pollard, D. (1984). Convergence of stochastic processes . Springer, New York

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Rand, W. M. (1971). Objective criteria for the evaluation of clustering methods. Journal of the American Statistical association , 66(336):846--850

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Steinhaus, H. (1956). Sur la division des corp materiels en parties. Bull. Acad. Polon. Sci , 1(804):801

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Yohai, V. J. and Zamar, R. H. (1988). High breakdown-point estimates of regression by means of the minimization of an efficient scale. Journal of the American statistical association , 83(402):406--413

work page 1988

[1] [1]

J., and Zamar, R

Agostinelli, C., Leung, A., Yohai, V. J., and Zamar, R. H. (2015). Robust estimation of multivariate location and scatter in the presence of cellwise and casewise contamination. Test , 24(3):441--461

work page 2015

[2] [2]

Al Hasan, M., Chaoji, V., Salem, S., and Zaki, M. J. (2009). Robust partitional clustering by outlier and density insensitive seeding. Pattern Recognition Letters , 30(11):994--1002

work page 2009

[3] [3]

H., Sun, Y., and Wang, J

Cheng, H.-D., Jiang, X. H., Sun, Y., and Wang, J. (2001). Color image segmentation: advances and prospects. Pattern recognition , 34(12):2259--2281

work page 2001

[4] [4]

Cuesta-Albertos, J., Gordaliza, A., and Matr \'a n, C. (1997). Trimmed k -means: An attempt to robustify quantizers. The Annals of Statistics , 25(2):553--576

work page 1997

[5] [5]

A., and Mayo-Iscar, A

Fritz, H., Garc a-Escudero, L. A., and Mayo-Iscar, A. (2012). tclust: An r package for a trimming approach to cluster analysis. Journal of Statistical Software , 47(12):1--26

work page 2012

[6] [6]

A., Gordaliza, A., Matr \'a n, C., and Mayo-Iscar, A

Garc \' a-Escudero, L. A., Gordaliza, A., Matr \'a n, C., and Mayo-Iscar, A. (2008). A general trimming approach to robust cluster analysis. The Annals of Statistics , pages 1324--1345

work page 2008

[7] [7]

Hartigan, J. A. and Wong, M. A. (1979). Algorithm as 136: A k-means clustering algorithm. Journal of the Royal Statistical Society. Series C (Applied Statistics) , 28(1):100--108

work page 1979

[8] [8]

Leung, A., Danilov, M., Yohai, V., and Zamar, R. (2015). Gse: Robust estimation in the presence of cellwise and casewise contamination and missing data. Test , page R package

work page 2015

[9] [9]

Lloyd, S. (1982). Least squares quantization in pcm. IEEE transactions on information theory , 28(2):129--137

work page 1982

[10] [10]

MacQueen, J. (1967). Some methods for classification and analysis of multivariate observations. In Proceedings of the fifth Berkeley symposium on mathematical statistics and probability , number 14, pages 281--297. Oakland, CA, USA

work page 1967

[11] [11]

Maronna, R. A. and Yohai, V. J. (2017). Robust and efficient estimation of multivariate scatter and location. Computational Statistics & Data Analysis , 109:64--75

work page 2017

[12] [12]

Munkres, J. (2000). Topology . Prentice Hall, Upper Saddle River, NJ

work page 2000

[13] [13]

View of Curiosity's First Scoop Also Shows Bright Object

NASA (2016). View of Curiosity's First Scoop Also Shows Bright Object

work page 2016

[14] [14]

Pollard, D. (1981). Strong consistency of k -means clustering. The Annals of Statistics , 9(1):135--140

work page 1981

[15] [15]

Pollard, D. (1984). Convergence of stochastic processes . Springer, New York

work page 1984

[16] [16]

Rand, W. M. (1971). Objective criteria for the evaluation of clustering methods. Journal of the American Statistical association , 66(336):846--850

work page 1971

[17] [17]

Steinhaus, H. (1956). Sur la division des corp materiels en parties. Bull. Acad. Polon. Sci , 1(804):801

work page 1956

[18] [18]

Yohai, V. J. and Zamar, R. H. (1988). High breakdown-point estimates of regression by means of the minimization of an efficient scale. Journal of the American statistical association , 83(402):406--413

work page 1988