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REVIEW 3 major objections 6 minor 28 references

Adding inter-cluster distance to the k-means objective produces more accurate clusters and better outlier placement.

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2026-08-03 20:52 UTC pith:IJORGWUA

load-bearing objection The paper's central mechanism is missing—Algorithm 1 never uses the inter-cluster distance in updates—so the reported gains are unsupported. the 3 major comments →

arxiv 2511.17823 v2 pith:IJORGWUA submitted 2025-11-21 cs.LG cs.CEstat.MEstat.ML

A novel k-means clustering approach using two distance measures for Gaussian data

Naitik Gada (1) ((1) Rochester Institute of Technology) This is my paper
classification cs.LG cs.CEstat.MEstat.ML
keywords k-means clusteringinter-cluster distancewithin-cluster distanceGaussian dataoutlier clusteringcluster validity indexunsupervised learningbenchmark data sets
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

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

This paper argues that a k-means variant using two distance measures — the within-cluster distance (how tightly points sit around their center) and the inter-cluster distance (how far apart different clusters are) — clusters Gaussian data more accurately than classic k-means. The proposed algorithm sets the number of clusters with a standard validity index, then optimizes a relative cost that shrinks within-cluster distance while growing between-cluster distance. Across four synthetic Gaussian data sets (2D and 3D, two variances) and three benchmark data sets, the paper reports higher overall accuracy, recall, precision, and F1 for the two-distance method, along with more consistent results across random initializations and better placement of outlier points. If the claim holds, a simple change to k-means' objective could make clustering more reliable in settings where data is roughly Gaussian and outliers matter.

Core claim

The central claim is that including both within-cluster distance (WCD) and inter-cluster distance (ICD) in the k-means objective produces better clusters than the traditional within-cluster-only objective. WCD is defined as the sum, over clusters, of squared Euclidean distances from each point to its cluster center; ICD is the summed distance between points in different clusters. The algorithm first determines k using a cluster-validity index, initializes centers, then loops through assignment and centroid-recomputation while aiming to optimize the relative cost C_j/D — the ratio of within-cluster cost to inter-cluster distance. On the paper's evidence, this ratio makes the algorithm more ac

What carries the argument

The load-bearing object is the relative cost C_j/D: the ratio of the within-cluster squared distance summed over clusters (C_j) to the inter-cluster distance (D), which sums distances between points in different clusters. The paper's Algorithm 1 wraps this ratio around the usual k-means loop — assign each point to the nearest center, recompute each center as the mean of its cluster — and reports that optimizing this ratio is what improves convergence, outlier assignment, and stability to initial centers. A cluster-validity index fixes k before the loop runs.

Load-bearing premise

The load-bearing premise is that the pseudocode step 'Optimize relative cost, C_j/D' actually changes how points are assigned to clusters or how centroids move; the paper does not give the equation linking ICD to the assignment or update, and if that step is a no-op, the reported accuracy gains cannot be attributed to including inter-cluster distance.

What would settle it

Instrument Algorithm 1 at the 'Optimize relative cost' step: log cluster assignments and centroids before and after evaluating C_j/D, then rerun the same code with the ICD term (D) artificially held constant. If assignments and centroids are identical in both runs, the ICD term is not doing any work and the reported gains come from elsewhere; if they differ, the two-distance mechanism is real.

Watch this falsifier — get emailed when new claim-graph text bears on it.

If this is right

  • A simple change to the k-means objective — adding an inter-cluster separation term — can yield higher clustering accuracy without adding user-tuned hyperparameters.
  • Outlier points near cluster boundaries, which classic k-means tends to misassign, are placed into their true cluster more often when between-cluster separation is part of the objective.
  • The method's accuracy fluctuates less across random initial centroid placements, so it partially addresses k-means' well-known initialization sensitivity.
  • The improvement persists on overlapping, higher-variance data, indicating the method is not limited to well-separated clusters.

Where Pith is reading between the lines

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

  • Editorial inference: Since the paper never writes an explicit assignment or update equation for the ICD term, a fair test of the mechanism would be to vary D in the 'Optimize relative cost' step and check whether cluster assignments change; without such a check, the reported gains could be an artifact of implementation details rather than the two-distance objective.
  • Editorial inference: The average-over-100-runs design with identical initial centers for both methods means part of the reported advantage may reflect reduced variance across restarts rather than a direct ICD effect; an ablation that fixes the assignment rule exactly would isolate the cause.
  • Editorial inference: High-dimensional results show only a small gain on the breast-cancer set, so the method's advantage may shrink as dimensionality grows; testing on a range of dimensionalities would map where the ratio helps.
  • Editorial inference: A natural next step is to derive a joint assignment rule — assign each point to the cluster that minimizes the ratio C_j/D — and compare it against stronger initialization baselines to see whether ICD still helps once initialization is controlled.

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

3 major / 6 minor

Summary. The paper proposes a k-means variant that, in addition to within-cluster distance (WCD), uses inter-cluster distance (ICD) to form clusters. The algorithm selects k via the Calinski-Harabasz criterion, initializes centroids randomly, and claims to optimize a relative cost C_j/D. Experiments on four synthetic Gaussian datasets and three UCI benchmarks report modest improvements in accuracy, recall, precision, and F1 over traditional k-means, plus better visual outlier handling. The central contribution is the joint use of WCD and ICD.

Significance. If substantiated, explicitly maximizing inter-cluster separation within k-means is a reasonable direction and could be a small incremental contribution. The paper sensibly reports standard metrics and uses shared initialization for comparison. However, the manuscript provides no concrete optimization step through which ICD affects cluster assignment or centroid updates; the described procedure is standard k-means. Consequently, the reported improvements are not attributable to the proposed method. The evaluation also lacks error bars and uses a binary confusion matrix for multiclass problems. As it stands, the central claim is unsupported.

major comments (3)
  1. [Section 3.2/3.3, Algorithm 1] The pseudocode step 'Optimize relative cost, C_j/D' (Algorithm 1, line 7) is never operationalized. Section 3.1 describes only nearest-centroid assignment and mean recomputation, which minimize the within-cluster sum of squares (sum of C_j), not the ratio sum C_j / D. No equation in Sections 3.2 or 3.3 shows how D enters the assignment rule or the centroid update. As written, the algorithm is identical to vanilla k-means, so the accuracy differences in Tables 5.1–5.7 cannot be attributed to ICD. The authors must specify the exact objective and the update rules that use D, and ideally prove or demonstrate that the relative-cost objective is decreased by those updates.
  2. [Section 5, Tables 5.1–5.7] The paper states (e.g., p. 15) that both methods use the same initial centroid positions for each iteration. Under identical initialization and the identical updates described in Section 3.1, the two methods would produce identical clusterings; the reported differences imply either an undisclosed implementation difference or an evaluation artifact. Moreover, no standard deviations or significance tests accompany the averages over 100 iterations. On Breast Cancer the difference is only 0.3%, and Figure 5.17 shows the traditional method frequently scoring higher. The claim of consistent outperformance is therefore not established by the reported statistics.
  3. [Section 3.2, confusion matrix] The definitions of TP/FN/FP/TN and overall accuracy are for binary classification, but the experiments include three-class datasets (Iris, Wine) and the synthetic data have k=3. The paper does not explain how the 2x2 confusion matrix is extended to multiclass problems, nor how cluster labels are matched to ground-truth labels before accuracy is computed. Without this, the accuracy, recall, precision, and F1 numbers in all multiclass tables are not well-defined.
minor comments (6)
  1. [Section 3.1] The phrase 'the average of the sum of the within-cluster distance' is unclear; presumably the new centroid is the mean of the points in the cluster, not an average of a sum. Please rewrite.
  2. [Algorithm 1] The loop variables M and Z are not defined, and the two nested loops are both named 'iteration', which is confusing. The role of the inner loop relative to the outer loop is not explained.
  3. [Table 5.7] The table header contains a typo: 'Breast Caner data set' should be 'Breast Cancer data set'.
  4. [Section 5.2.1 and 5.2.2] Assertions about specific outlier points (e.g., near (2,2,0) in Iris, or around (0.6,0.5,2.5) in Wine) are based on visual inspection only. No quantitative measure of outlier-clustering performance is provided, so this claim remains anecdotal.
  5. [Section 4.1] The synthetic data generation via Kronecker product is described only verbally. The exact offsets, sample sizes, and how the Kronecker product is used are not specified, which hinders reproducibility.
  6. [References] Several references are incomplete or inconsistent, e.g., reference [5] lacks author names, and [23] is a Wikipedia citation. Please standardize the bibliography.

Circularity Check

0 steps flagged

No material circularity: the empirical claim is not definitionally loaded, but the ICD term is never wired into the algorithm (an omitted-proof/correctness issue), and the one self-citation is unrelated and non-load-bearing.

full rationale

The WCD/ICD definitions are independent of the benchmark labels, and the reported accuracy is computed from a confusion-matrix comparison against external labels, so no fitted parameter is relabeled as a prediction and no quantity is defined in terms of the result it is supposed to support. The only self-citation ([28], an aluminum-specifications paper) appears in a literature list about initialization methods and is not load-bearing. The most serious defect is not circular: Algorithm 1's 'Optimize relative cost, C_j/D' (line 7) is never connected to the nearest-centroid assignment or the centroid update; Section 3.1 and Algorithm 1 (lines 8-11) describe only standard k-means mean updates. If the implementation matches the pseudocode, the method is vanilla k-means and the reported differences are unexplained. That is an omitted proof/empirical validity problem, not a definitional reduction: the conclusion is not assumed in the premises, it is simply not derived. Accordingly, no circularity step is reported and the score reflects only a minor non-load-bearing self-citation.

Axiom & Free-Parameter Ledger

1 free parameters · 4 axioms · 0 invented entities

The central claim rests on an undefined relationship between the relative-cost objective and the k-means update. No free parameter is fitted to the benchmark labels except the choice of k; no new entity is introduced.

free parameters (1)
  • Number of clusters k = 3 for synthetic/Iris/Wine; 2 for Breast Cancer
    Determined by Calinski-Harabasz index and elbow method; a modeling choice that affects all reported accuracies and is not fit to test labels.
axioms (4)
  • standard math Standard k-means mean-update converges and minimizes within-cluster sum of squares
    Invoked in Sections 3.1 and 3.3 as the basis for the update; not proved in the paper.
  • ad hoc to paper The relative cost C_j/D is optimized by nearest-centroid assignment plus mean recomputation
    Required for the claimed method to differ from k-means; no derivation is given in Section 3.3.
  • domain assumption Calinski-Harabasz index recovers the true number of clusters for these datasets
    Algorithm step 1 uses CH and elbow to fix k; no validation is provided.
  • domain assumption Ground-truth label accuracy is the appropriate measure of clustering quality
    Used to compute OA, precision, recall and F1 in Section 3.2; a standard but contested choice.

pith-pipeline@v1.3.0-alltime-deepseek · 13870 in / 10443 out tokens · 93068 ms · 2026-08-03T20:52:04.320724+00:00 · methodology

0 comments
read the original abstract

Clustering algorithms have long been the topic of research, representing the more popular side of unsupervised learning. Since clustering analysis is one of the best ways to find some clarity and structure within raw data, this paper explores a novel approach to k-means clustering. Here we present a k-means clustering algorithm that takes both the within cluster distance (WCD) and the inter cluster distance (ICD) as the distance metric to cluster the data into k clusters pre-determined by the Calinski-Harabasz criterion in order to provide a more robust output for the clustering analysis. The idea with this approach is that by including both the measurement metrics, the convergence of the data into their clusters becomes solidified and more robust. We run the algorithm with some synthetically produced data and also some benchmark data sets obtained from the UCI repository. The results show that the convergence of the data into their respective clusters is more accurate by using both WCD and ICD measurement metrics. The algorithm is also better at clustering the outliers into their true clusters as opposed to the traditional k means method. We also address some interesting possible research topics that reveal themselves as we answer the questions we initially set out to address.

Figures

Figures reproduced from arXiv: 2511.17823 by Naitik Gada (1) ((1) Rochester Institute of Technology).

Figure 5.1
Figure 5.1. Figure 5.1: 2-dimensional data with variance = 0.5 As seen in figure 5.1, the clusters are segregated enough which makes them easy to identify since they are closely packed. The accuracy results of this data set are summarized in table 5.1. 13 [PITH_FULL_IMAGE:figures/full_fig_p014_5_1.png] view at source ↗
Figure 5.2
Figure 5.2. Figure 5.2: Overall Accuracy of the proposed k-means vs traditional k-means over 100 iterations (variance = 0.5) We can see what the accuracy for each method looks like over 100 iterations in figure 5.2. The faint blue line represents the new proposed algorithm while the orange line represents the traditional k-means algorithm. It’s clear that the results vary with every iteration with only a few consistent accuracy… view at source ↗
Figure 5.3
Figure 5.3. Figure 5.3: 2-dimensional data with variance = 1 [PITH_FULL_IMAGE:figures/full_fig_p016_5_3.png] view at source ↗
Figure 5.4
Figure 5.4. Figure 5.4: Overall Accuracy of the proposed k-means vs traditional k-means over 100 iterations (variance = 1) We can see how often the traditional k-means drops in accuracy to sub-70 percent as compared to the new proposed algorithm. This might be a result of the cluster center initialization but the code was set up in a way that had the same initialization positions for both methods. The overall accuracy is higher… view at source ↗
Figure 5.5
Figure 5.5. Figure 5.5: 3-dimensional data with variance = 0.5 The Overall Accuracy is just above 97 percent with the proposed algorithm and just under 93 percent for the traditional k-means algorithm. The proposed algorithm outperforms the traditional method by about 4 percent. The Recall, Precision and F Score values are all tabulated in table 5.3. You can see the comparative per￾formance of both methods in figure 5.6. The in… view at source ↗
Figure 5.6
Figure 5.6. Figure 5.6: Overall Accuracy of the proposed k-means vs traditional k-means over 100 iterations (variance = 0.5) 3-dimensional, Variance = 0.5 Algorithm Accuracy Recall Precision F Score Proposed k-means 0.9706 0.9566 0.9619 0.9591 Traditional k￾means 0.9298 0.9155 0.9239 0.9195 [PITH_FULL_IMAGE:figures/full_fig_p019_5_6.png] view at source ↗
Figure 5.7
Figure 5.7. Figure 5.7: 3-dimensional data with variance = 1 3-dimensional, Variance = 1 Algorithm Accuracy Recall Precision F Score Proposed k-means 0.9743 0.9755 0.9636 0.9695 Traditional k￾means 0.9514 0.9466 0.9422 0.9443 [PITH_FULL_IMAGE:figures/full_fig_p020_5_7.png] view at source ↗
Figure 5.8
Figure 5.8. Figure 5.8: Overall Accuracy of the proposed k-means vs traditional k-means over 100 iterations (variance = 1) 20 [PITH_FULL_IMAGE:figures/full_fig_p021_5_8.png] view at source ↗
Figure 5.9
Figure 5.9. Figure 5.9: The raw Iris data clusters Running both the proposed algorithm and the traditional k-means algorithm, we can see that the proposed algorithm outperforms the traditional method across the board [PITH_FULL_IMAGE:figures/full_fig_p022_5_9.png] view at source ↗
Figure 5.10
Figure 5.10. Figure 5.10: Overall Accuracy for Iris What is interesting to note is that even though the drops in accuracy at certain intervals align almost perfectly between the two methods, the maximum accuracy achieved through the proposed algorithm is higher than the maximum accuracy with the traditional method. The proposed algorithm reaches a maximum accuracy of about 95.33 percent while the traditional method cannot break … view at source ↗
Figure 5.11
Figure 5.11. Figure 5.11: Clusters produced by the proposed method vs traditional [PITH_FULL_IMAGE:figures/full_fig_p024_5_11.png] view at source ↗
Figure 5.12
Figure 5.12. Figure 5.12: The raw Wine data clusters Results of the wine data set are summarized in table 5.6. It is important to note that the scatter plot shown in figure 5.12 only shows 3 of the 13 dimensions or attributes of the entire data set. The proposed algorithm resulted in a higher overall accuracy along with recall, precision and F scores for the data as compared to the traditional method. Although the accuracy is no… view at source ↗
Figure 5.13
Figure 5.13. Figure 5.13: Overall Accuracy for Wine [PITH_FULL_IMAGE:figures/full_fig_p026_5_13.png] view at source ↗
Figure 5.14
Figure 5.14. Figure 5.14: Clusters produced by the proposed method vs traditional [PITH_FULL_IMAGE:figures/full_fig_p027_5_14.png] view at source ↗
Figure 5.15
Figure 5.15. Figure 5.15: The raw breast cancer data set clusters The breast cancer data set comprises of the digital information from images of breast mass, describing the characteristics of cell nuclei present in the image. The data set itself is a 9 dimensional data set with the last feature being the diagnosis (classification) of that data point (either Benign or Malignant). All the previous data sets we’ve used have only be… view at source ↗
Figure 5.16
Figure 5.16. Figure 5.16: Clusters produced by the proposed method vs traditional [PITH_FULL_IMAGE:figures/full_fig_p029_5_16.png] view at source ↗
Figure 5.17
Figure 5.17. Figure 5.17: Overall Accuracy for Breast Cancer We can see in figure 5.17 how the volatility compares between the two methods, iteration over iteration. We’ve seen this happen with the Iris and Wine data sets as well but with the breast cancer data set, it is more robust and apparent. While the proposed method consistently hovers in the 96.2 to 96 percent range, the traditional method jumps between 96.4 to 95.4 perc… view at source ↗

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