FastCentNN: Accelerating Centroid Neural Network with Entropy Proxy

FastCentNN: Accelerating Centroid Neural Network with Entropy Proxy Le-Anh Tran

arXiv:2607.13613v1 [cs.LG] 15 Jul 2026

[email protected]

Abstract—Centroid neural network (CentNN) is an unsupervised competitive learning algorithm in which centroid splitting is triggered only after strict local stabilization, often leading to prolonged low-movement training phases before model expansion. This report proposes FastCentNN, an accelerated variant that addresses this inefficiency by introducing an early splitting strategy based on the total centroid movement per epoch, which serves as a training entropy proxy. As a result, FastCentNN reduces unnecessary reassignment epochs while preserving the original winner-loser learning dynamics. FastCentNN supports both absolute and stage-relative movement thresholds, allowing the splitting criterion to remain either fixed or adaptive throughout training. Experiments on some benchmark datasets show that FastCentNN consistently achieves clustering quality comparable to CentNN while reducing runtime by up to 16% on synthetic 2D datasets and about 5% on high-dimensional datasets. FastCentNN therefore provides a practical and efficient drop-in replacement for CentNN, retaining its online adaptive learning behavior while offering a simple and interpretable speed-stability trade-off through configurable splitting thresholds. Code is available1 .

1. Introduction Clustering remains a core task in unsupervised learning, with applications spanning pattern discovery, compression, anomaly detection, and post-processing [1]–[6]. Among modern clustering approaches, centroid neural network (CentNN) [3] is a prototype-based algorithm that follows a competitive-learning paradigm, where centroid vectors are updated online through winner-loser dynamics. This design is well suited to incremental data processing, memory-constrained environments, and adaptive model growth, placing CentNN within the broader family of competitive and self-organizing neural models [7], [8]. Unlike batch clustering methods, CentNN updates centroids one sample at a time and expands model capacity through centroid splitting. Despite these advantages, standard CentNN suffers from a practical inefficiency: before each split event, it may spend many epochs in a low-activity regime where centroid movement is minimal, yet splitting is delayed until strict local stabilization, as illustrated in Fig. 1. These “flat” phases increase runtime without corresponding structural progress. To address this limitation, this report proposes 1. https://github.com/tranleanh/centroid-neural-networks

FastCentNN, a more efficient variant that treats global centroid movement per epoch as an entropy proxy for clustering uncertainty and initiates splitting once movement remains below a threshold for a patience window. FastCentNN converts split timing from a strict convergence event into an adaptive early-trigger decision while preserving the original winner-loser learning dynamics. It supports both absolute and stage-relative thresholds, enabling consistent behavior across datasets and training stages. Although the asymptotic computational complexity remains unchanged, FastCentNN improves practical efficiency by reducing unnecessary lowinformation epochs between split operations.

2. FastCentNN 2.1. Problem Setup Let X = {xi }ni=1 , with xi ∈ Rd , and let K be the target number of clusters. CentNN represents clusters by t at epoch t, where kt grows over time centroids {wj }kj=1 via split operations. For each sample x, the winner index is defined as: j ⋆ (x) = arg

min

j∈{1,...,kt }

∥x − wj ∥2 .

(1) (t−1)

Each sample also stores its previous assignment ai the previous epoch.

from

2.2. Winner-Loser Updates If sample xi changes assignment at epoch t from pre(t−1) vious cluster p = ai to current winner c = j ⋆ (xi ), the centroids are updated by: 1 wc ← wc + (xi − wc ), (2) Nc + 1 1 wp ← wp − (xi − wp ), (3) Np − 1 where Nc , Np are current member counts before reassignment. The loser update is applied only when Np > 1 to avoid division by zero and unstable singleton collapse.

2.3. Split Operation CentNN starts from two centroids near the global mean: n

w1 = x̄ + ϵ,

w2 = x̄ − ϵ,

x̄ =

1X xi , n i=1

(4)

CentNN Clustering (2D)

15

Centroid Movement Over Time Movement per epoch A typical low movement threshold

12 Total centroid movement (L2 sum)

10 5 0 5 10

10 8 6 4 2 0

10

5

0

5

10

15

0

10

20

30 40 Epoch index

50

60

Figure 1. Clustering result (left) and convergence behavior (right) of CentNN on a synthetic 2D dataset. CentNN converges through occasional large centroid adjustments after split events, separated by periods of minimal movement.

where ϵ is a small perturbation. The centroids are updated iteratively until local stabilization is reached. If kt < K , the split candidate is selected based on the maximum withincluster error: X s = arg max Ej , Ej = ∥xi − wj ∥2 . (5) j

xi :ai =j

Then a symmetric split is applied: ws− = ws − δs ,

ws+ = ws + δs ,

(6)

where δs is a small perturbation (magnitude controlled by ϵ), typically aligned with the largest-variance feature within cluster s.

2.4. Early Split in FastCentNN Standard CentNN often spends many epochs in a lowactivity regime with minimal centroid movement before satisfying the splitting criterion. FastCentNN addresses this inefficiency by replacing strict stabilization with an early splitting criterion based on the total centroid movement: Mt =

kt X

(t)

(t−1)

∥wj − wj

∥2 ,

(7)

Algorithm 1 Fast Centroid Neural Network (FastCentNN) Require: Dataset X , target num of clusters K , max epochs T , split scale ϵ, movement threshold τ , patience P 1: Initialize two centroids near the global mean 2: Initialize cluster assignments and baseline movement B 3: for t = 1 to T do 4: Perform one epoch of winner-loser centroid updates 5: Compute assignment-change count Ct 6: Compute total centroid movement Mt 7: Update patience counter using Mt and τt 8: split ← (patience ≥ P ) ∨ (Ct = 0) 9: if kt < K and split then 10: Pick the cluster with max within-cluster error 11: Split the selected cluster’s centroid using ϵ 12: Reset B and patience counter 13: else if Ct = 0 then 14: break 15: end if 16: end for 17: return final centroids and cluster assignments

where ε0 > 0 is a small constant for numerical stability. The full FastCentNN procedure is summarized in Algorithm 1.

j=1

where Mt is the global centroid movement (entropy proxy) at epoch t. A split is triggered when Mt remains below a threshold for a predefined patience window: Mt ≤ τt for P consecutive epochs,

streak ← 0,

3.1. Experimental Setup

(8)

where τt is the movement threshold and P is the patience parameter. The threshold can be configured in two modes. In the absolute mode, τt = τ , whereas in the stage-relative mode, τt = τ Br , where Br is the baseline movement measured immediately after the latest split stage r. Following each split, the stage state is reset as: Br+1 ← max(Mt , ε0 ),

3. Experiments

(9)

FastCentNN is evaluated against the original CentNN under identical parameter settings on two clustering tasks: (i) synthetic 2D datasets and (ii) high-dimensional datasets. Since the primary objective is to improve computational efficiency while preserving clustering quality, the relative difference in mean squared error (MSE) is reported as: ∆MSE(%) =

MSEFastCentNN − MSECentNN × 100. MSECentNN

(10)

TABLE 1. P ERFORMANCE OF {C ENT NN / FAST C ENT NN} ON SYNTHETIC 2D DATASETS . Dataset A1 A2 S1 S2 R15 Aggregation

Clusters 20 35 15 15 15 7

Instances 3,000 5,250 5,000 5,000 600 788

Runtime (s) 1.575 / 1.312 4.771 / 4.324 1.110 / 1.048 1.856 / 1.728 0.077 / 0.073 0.113 / 0.096

Epochs 155 / 128 274 / 246 64 / 61 110 / 102 34 / 33 40 / 34

∆MSE (%) +2.47×10−7 +5.18×10−7 0 +7.53×10−7 0 -2.83×10−2

Speed Up 16.72% 9.37% 5.54% 6.90% 4.18% 14.80%

TABLE 2. P ERFORMANCE OF {C ENT NN / FAST C ENT NN} ON MNIST AND FASHION -MNIST. Dataset MNIST Fashion-MNIST

Clusters 10 10

Instances 10,000 10,000

Runtime (s) 8.344 / 7.924 8.220 / 7.775

In addition, efficiency is assessed using the total runtime and the number of training epochs required for convergence.

3.2. Test on Synthetic 2D Data FastCentNN was evaluated against the original CentNN on six widely used synthetic clustering benchmark datasets: A1, A2, S1, S2, R15, and Aggregation [9]. Each dataset consists of 2D data points with a predefined number of clusters. Table 1 summarizes the results. Compared with the original CentNN, FastCentNN consistently reduces both runtime and the number of training epochs while producing nearly identical clustering quality. Across all six datasets, FastCentNN achieves an average runtime reduction of approximately 9.6%, with the largest speed-up of 16.72% on A1 and the smallest of 4.18% on R15. The ∆MSE measures are negligible for all datasets, remaining close to zero even on the most challenging cases. These results indicate that the proposed early splitting strategy effectively eliminates unnecessary low-movement epochs, improving computational efficiency without degrading clustering performance. Visual results are shown in Fig. 2.

3.3. Test on High-dimensional Data FastCentNN was also evaluated on two high-dimensional image datasets, MNIST and Fashion-MNIST, each configured with 10 target clusters. Table 2 summarizes the results. FastCentNN consistently reduces both runtime and the number of training epochs while preserving clustering quality. On MNIST, the runtime is improved from 8.344 s to 7.924 s, corresponding to a speed-up of 5.03%, while the number of training epochs is decreased from 141 to 132. Similarly, on Fashion-MNIST, FastCentNN reduces the runtime from 8.220 s to 7.775 s (5.41% speed-up) and the number of training epochs from 127 to 119. The ∆MSE values are negligible on both datasets, remaining within 10−5 % compared to the original CentNN. These results demonstrate that the proposed early splitting strategy remains effective on high-dimensional datasets, reducing computational cost while maintaining overall clustering accuracy.

Epochs 141 / 132 127 / 119

∆MSE (%) +1.53×10−5 -6.37×10−5

Speed Up 5.03% 5.41%

4. Conclusion This report proposes FastCentNN, an accelerated variant of CentNN that introduces an early splitting strategy based on total centroid movement. By triggering centroid splits before strict local stabilization, FastCentNN reduces unnecessary low-movement epochs while preserving the original winner-loser learning dynamics. Experimental results on synthetic 2D datasets, MNIST, and Fashion-MNIST demonstrate consistent reductions in runtime and training epochs with negligible differences in clustering quality.

Acknowledgment The author would like to thank Prof. Dong-Chul Park, the inventor of the original CentNN algorithm, whose insights inspired the development of this approach.

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Figure 2. Comparison on synthetic 2D datasets.