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

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.

Silhouette Aggregation: From Micro to Macro

Silhouette coefficient is an established internal clustering evaluation measure that produces a score per data point, assessing the quality of its clustering assignment. To assess the quality of the clustering of the whole dataset, the scores of all the points in the dataset are either (micro) averaged into a single value or averaged at the cluster level and then (macro) averaged. As we illustrate in this work, by using a synthetic example, the micro-averaging strategy is sensitive both to cluster imbalance and outliers (background noise) while macro-averaging is far more robust to both. Furthermore, the latter allows cluster-balanced sampling which yields robust computation of the silhouette score. By conducting an experimental study on eight real-world datasets, estimating the ground truth number of clusters, we show that both coefficients, micro and macro, should be considered.

UniForCE: The Unimodality Forest Method for Clustering and Estimation of the Number of Clusters

Estimating the number of clusters k while clustering the data is a challenging task. An incorrect cluster assumption indicates that the number of clusters k gets wrongly estimated. Consequently, the model fitting becomes less important. In this work, we focus on the concept of unimodality and propose a flexible cluster definition called locally unimodal cluster. A locally unimodal cluster extends for as long as unimodality is locally preserved across pairs of subclusters of the data. Then, we propose the UniForCE method for locally unimodal clustering. The method starts with an initial overclustering of the data and relies on the unimodality graph that connects subclusters forming unimodal pairs. Such pairs are identified using an appropriate statistical test. UniForCE identifies maximal locally unimodal clusters by computing a spanning forest in the unimodality graph. Experimental results on both real and synthetic datasets illustrate that the proposed methodology is particularly flexible and robust in discovering regular and highly complex cluster shapes. Most importantly, it automatically provides an adequate estimation of the number of clusters.

Global $k$-means$++$: an effective relaxation of the global $k$-means clustering algorithm

The $k$-means algorithm is a very prevalent clustering method because of its simplicity, effectiveness, and speed, but its main disadvantage is its high sensitivity to the initial positions of the cluster centers. The global $k$-means is a deterministic algorithm proposed to tackle the random initialization problem of k-means but requires high computational cost. It partitions the data to $K$ clusters by solving all $k$-means sub-problems incrementally for $k=1,\ldots, K$. For each $k$ cluster problem, the method executes the $k$-means algorithm $N$ times, where $N$ is the number of data points. In this paper, we propose the global $k$-means$++$ clustering algorithm, which is an effective way of acquiring quality clustering solutions akin to those of global $k$-means with a reduced computational load. This is achieved by exploiting the center section probability that is used in the effective $k$-means$++$ algorithm. The proposed method has been tested and compared in various well-known real and synthetic datasets yielding very satisfactory results in terms of clustering quality and execution speed.