Automatic depth-based local center clustering via $β$-integrated local depth and adaptive grouping

This paper introduces a semi-supervised clustering framework grounded in the statistical duality between grouping principles and anomaly detection. We address the challenge of robust cluster definition in noisy environments -- a task where partitioning algorithms often over-assign outliers and density-based methods remain sensitive to heuristic global parameters. Drawing on \textit{a-contrario} statistical reasoning and Gestalt proximity principles, we define a cluster as a maximal subset of data points containing no anomalies relative to a null hypothesis of uniform randomness. Central to this approach is the Perception algorithm, which utilises a principled expectation-based threshold ($\mathbb{E} < 1$) to identify outliers without manual parameter tuning. By treating clustering as the dual of anomaly detection, we employ an iterative ``clustering-by-exclusion'' mechanism. The algorithm is seed-guided, leveraging minimal user-provided labels to initialise robust cluster medians and form initial groups, which are subsequently expanded by admitting non-anomalous points. This approach naturally isolates fringe points, isolated noise, and emerging unknown clusters. We evaluate the method on synthetic and real-world benchmarks, including image and text datasets represented through raw, linear-reduced, and neighbourhood-preserving embeddings. Results demonstrate that with as few as 10--30 seeds per cluster, the proposed method achieves competitive and often very strong performance under a practical low-tuning benchmarking protocol, while maintaining linear scalability with respect to both observations and dimensionality for a fixed number of seeded clusters and iterations.

Clustering is a cornerstone of modern data analysis. Detecting clusters in exploratory data analyses (EDA) requires algorithms that make few assumptions about the data. Density-based clustering algorithms are particularly well-suited for EDA because they describe high-density regions, assuming only that a density exists. Applying density-based clustering algorithms in practice, however, requires selecting appropriate hyperparameters, which is difficult without prior knowledge of the data distribution. For example, DBSCAN requires selecting a density threshold, and HDBSCAN* relies on a minimum cluster size parameter. In this work, we propose Persistent Leaves Spatial Clustering for Applications with Noise (PLSCAN), a multiscale density-based clustering algorithm that replaces HDBSCAN*'s fixed minimum cluster size pruning of a mutual-reachability linkage hierarchy with a persistence-based cluster selection procedure. Effectively, PLSCAN identifies all minimum cluster sizes for which HDBSCAN* produces stable (leaf) clusters. In concept, PLSCAN applies scale-space clustering principles and is equivalent to persistent homology on a novel metric space. We compare its performance to HDBSCAN* on several real-world datasets, demonstrating that it achieves a higher median ARI, is less sensitive to changes in the number of mutual reachability neighbours, and has higher stability under resampling. Additionally, we compare PLSCAN's computational costs to $k$-Means++, demonstrating competitive run-times on low-dimensional datasets. At higher dimensions, run times scale more similarly to HDBSCAN*.

Clustering is a fundamental data mining technique for pattern recognition through unsupervised learning. Among various clustering methods, hierarchical clustering, density-based clustering, and graph clustering stand out as representative approaches. For hierarchical clustering, it can be categorized into agglomerative and divisive modes to construct clusters in a recursive manner. The key aspect of both modes is the calculation of inter-cluster similarity, which determines whether to merge the sub-clusters into one cluster or divide a current cluster into sub-clusters. Traditionally, the similarity is derived from pairwise distances, often overlooking density variations and structural connectivity in graphs. To address this, we propose a density-aware hierarchical clustering method based on element-categorized connection subgraphs (DHC-ECS), which effectively integrates the hierarchical clustering, density-based clustering, and graph clustering. Particularly, a novel inter-cluster similarity metric is introduced that considers not only distances but also the element categorization in the KNN connection subgraphs, kernel density estimation, and local connectivity within sub-clusters. Extensive evaluations on heterogeneous benchmark datasets demonstrate that DHC-ECS exhibits superior overall performance in terms of clustering accuracy and parameter robustness compared with the baseline methods (including AChameleon, RNN-DBSCAN, McDPC, and G-RMS). The work indicates the great potential of the proposed clustering algorithm for low-dimensional datasets by leveraging local density and graph-structured connectivity (i.e., the duality of vertices and edges), as well as the possibility to determine an intrinsic threshold, reducing the reliance on manual parameter tuning.