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Deep Clustering Using the Soft Silhouette Score: Towards Compact and Well-Separated Clusters

Georgios Vardakas, Ioannis Papakostas, Aristidis Likas

TL;DR

This work introduces the soft silhouette score, a differentiable probabilistic generalization of the classic silhouette, as a clustering objective for deep learning. It integrates soft silhouette into an autoencoder based architecture (DCSS) with a dedicated clustering head to produce cluster probabilities, jointly optimizing reconstruction and a clustering loss that promotes both compactness and inter-cluster separation. Empirical results across diverse real and synthetic datasets show that DCSS yields superior internal clustering quality (NMI, ARI) compared to established deep clustering methods, highlighting the benefit of optimizing cluster separability alongside compactness. The approach advances unsupervised deep clustering by providing a differentiable, probability-aware objective that improves representation learning for more distinct and cohesive clusters, with future work on data augmentation and model enhancements.

Abstract

Unsupervised learning has gained prominence in the big data era, offering a means to extract valuable insights from unlabeled datasets. Deep clustering has emerged as an important unsupervised category, aiming to exploit the non-linear mapping capabilities of neural networks in order to enhance clustering performance. The majority of deep clustering literature focuses on minimizing the inner-cluster variability in some embedded space while keeping the learned representation consistent with the original high-dimensional dataset. In this work, we propose soft silhoutte, a probabilistic formulation of the silhouette coefficient. Soft silhouette rewards compact and distinctly separated clustering solutions like the conventional silhouette coefficient. When optimized within a deep clustering framework, soft silhouette guides the learned representations towards forming compact and well-separated clusters. In addition, we introduce an autoencoder-based deep learning architecture that is suitable for optimizing the soft silhouette objective function. The proposed deep clustering method has been tested and compared with several well-studied deep clustering methods on various benchmark datasets, yielding very satisfactory clustering results.

Deep Clustering Using the Soft Silhouette Score: Towards Compact and Well-Separated Clusters

TL;DR

This work introduces the soft silhouette score, a differentiable probabilistic generalization of the classic silhouette, as a clustering objective for deep learning. It integrates soft silhouette into an autoencoder based architecture (DCSS) with a dedicated clustering head to produce cluster probabilities, jointly optimizing reconstruction and a clustering loss that promotes both compactness and inter-cluster separation. Empirical results across diverse real and synthetic datasets show that DCSS yields superior internal clustering quality (NMI, ARI) compared to established deep clustering methods, highlighting the benefit of optimizing cluster separability alongside compactness. The approach advances unsupervised deep clustering by providing a differentiable, probability-aware objective that improves representation learning for more distinct and cohesive clusters, with future work on data augmentation and model enhancements.

Abstract

Unsupervised learning has gained prominence in the big data era, offering a means to extract valuable insights from unlabeled datasets. Deep clustering has emerged as an important unsupervised category, aiming to exploit the non-linear mapping capabilities of neural networks in order to enhance clustering performance. The majority of deep clustering literature focuses on minimizing the inner-cluster variability in some embedded space while keeping the learned representation consistent with the original high-dimensional dataset. In this work, we propose soft silhoutte, a probabilistic formulation of the silhouette coefficient. Soft silhouette rewards compact and distinctly separated clustering solutions like the conventional silhouette coefficient. When optimized within a deep clustering framework, soft silhouette guides the learned representations towards forming compact and well-separated clusters. In addition, we introduce an autoencoder-based deep learning architecture that is suitable for optimizing the soft silhouette objective function. The proposed deep clustering method has been tested and compared with several well-studied deep clustering methods on various benchmark datasets, yielding very satisfactory clustering results.
Paper Structure (13 sections, 21 equations, 3 figures, 2 tables, 1 algorithm)

This paper contains 13 sections, 21 equations, 3 figures, 2 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Deep Clustering using Autoencoders
  3. The soft silhouette score
  4. Silhouette
  5. Soft Silhouette
  6. The DCSS method: Deep Clustering using Soft Silhouette
  7. Experiments
  8. Synthetic Data Demonstration
  9. Datasets
  10. Neural Network Architectures
  11. Evaluation
  12. Experimental Setup and Results
  13. Conclusions

Figures (3)

  • Figure 1: The proposed model architecture.
  • Figure 2: Synthetic demonstration of the representation learning capabilities of several methods. The generated 2-d dataset (top left) is hidden from the methods. Each method receives as input a 100-d dataset generated by non-linear transformations applied to the original 2-d data and provides a 2-d latent representation of the 100-d dataset, which is presented in the plots. Color indicates the true cluster labels.
  • Figure 3: Image clustering results on various datasets. In each sub-figure, rows correspond to different clusters. In each row the images are presented from left to right with decreasing cluster membership probability.