Cluster Number Estimation

Federated clustering methods that do not require the global number of clusters $K$ still assume that each client knows its local number $K_g$. This assumption is hard to justify when clients know no more about their data than the server does, as in fault diagnosis across independently operated industrial sites. We propose a two-phase framework in which neither count is known: each client first estimates $K_g$ from its own data, and an aggregator that requires local counts, such as FedGEM, then uses these estimates in place of the true values. For the first phase we introduce Adaptive Split--Merge (ASM), which grows a spherical Gaussian mixture by BIC-driven splitting and then merges excess components. ASM uses no labels, selects its hyperparameters on held-out client data only, and makes no assumption about how clusters are shared across clients. We derive a closed-form split criterion whose critical cluster size falls with anisotropy and rises with dimension, and show empirically that over-fragmentation grows with the number of points per cluster, which federation divides among clients. Across eight datasets, ASM with FedGEM attains a mean ARI of 0.333, against 0.256 for the next best label-free estimator and 0.361 when the true local counts are supplied. It also gives the most reliable global estimates of $K$ and is robust when client size is decoupled from local cardinality.

Multi-class anomaly classification in industrial vision remains challenging due to noisy/incomplete anomaly representations and the unknown number of anomaly classes. To overcome this, we propose MACO, a novel multi-class anomaly classification framework that learns comprehensive representations and dynamically estimates class number without prior knowledge. First, a soft-focus attention uses anomaly maps to concentrate on relevant abnormal regions, while suppressing background noise. Second, auxiliary classification ([A-CLS]) tokens complement the [CLS] token. They collectively attend to diverse anomaly sub-regions, yielding more holistic and discriminative features. These [A-CLS] tokens are also effective across more tasks and domains. To infer the class number, we propose Correlation-based Number Estimation strategy. It computes the average correlation among labeled classes and transfers its separability cue to the unlabeled set. Experiments on MVTec AD and MTD datasets demonstrate our superiority. Under known class number, MACO improves ARI by 6.5% and \textbf{16.3%} on both datasets, respectively. In the more challenging unknown number scenario, it achieves an \textbf{11.2%} NMI gain on MTD and outperforms existing number estimation strategies by \textbf{24.1%} UPS on MVTec AD. Code will be released at https://github.com/HUST-SLOW/MACO.

Clustering algorithms that operate on nearest-neighbor graphs, such as FINCH (First Integer Neighbor Clustering Hierarchy), depend heavily on the quality of the embedding space they are given. However, pretrained vision and language model embeddings are not optimized for this purpose. We propose CAESAR, a method that reorganizes a pretrained embedding space: a reorganization network is trained to pull mutual nearest neighbors together and push non-neighbors apart, yielding a reorganized embedding space substantially better suited to clustering. CAESAR offers a second major advantage: it never requires the number of clusters $K$. This matters because in realistic unsupervised settings, $K$ is typically unknown and discovering it is often part of the problem, yet most strong clustering methods take it as an input. We therefore design the entire CAESAR pipeline to infer $K$ rather than assume it is known. Empirically, reorganizing the embeddings consistently improves clustering over the raw space on both text and image datasets. Since the few deep clustering methods that also infer $K$ do not release their code, we complement controlled comparisons with methods that infer $K$ on the same embedding space by comparisons with strong deep clustering methods that are given the true $K$, giving them a substantial oracle advantage. Even so, CAESAR outperforms all of them on text, achieves the best results on the most challenging image benchmark and remains competitive on the others. Reorganizing pretrained embeddings thus emerges as a simple and powerful route to clustering realistic data, where classes overlap and the number of clusters is unknown.

The manifold hypothesis suggests a natural criterion for clustering: partition data according to the manifold component from which each point is drawn. Whether two components are separable depends on a geometric tradeoff: the ambient separation between components versus the largest gap in sampling. In practice, this tradeoff is rarely assessed explicitly, leading standard methods to over-commit to a single clustering assignment even when the data do not support a unique answer. We formalize this tradeoff by combining intrinsic manifold geometry (volume growth and reach) with sample-level quantities (fill distance and density), yielding a threshold phenomenon for mutual-$k$-nearest-neighbor graphs: when the offset-to-fill ratio exceeds a conservative upper threshold, component separation is preserved; below a lower threshold, components fuse. The gap between these thresholds defines a geometric uncertainty zone in which the number of clusters is not identifiable from the data. Nevertheless, conventional approaches still seek one: sweeping parameters (an engineering approach) or fitting a generative mixture model (a model-based approach). Rather than forcing a single estimate of the number of clusters, we propose Manifold-Based Clustering (MBC), which returns an explicit bracket interval to quantify the underlying data uncertainty. This bracket acts as an empirically calibrated diagnostic: it narrows when a single resolution is supported, widens when multiple resolutions coexist, and collapses to one when no separated structure is detectable. Empirically, we find that many real datasets lie within the uncertainty zone rather than admitting one clear answer. Our results suggest that ambiguity in cluster number is often intrinsic, and should be quantified rather than resolved.

Community detection is a fundamental problem in the analysis of complex networks. It has applications across social, biological, and financial domains. Traditional algorithms such as Louvain, LPA, and modularity optimization often require manual parameter tuning. They also suffer from inaccurate cluster center selection and struggle with scalability. To address these challenges, we propose Automatic Laplacian Centrality Means (ALCMeans), a novel community detection algorithm. ALCMeans combines Laplacian energy-based automatic center identification with DeepWalk embeddings for robust node representation. Unlike existing Laplacian-based and clustering methods, ALCMeans eliminates the need to predefine the number of communities, enhances cluster center selection using structural importance, and leverages representation learning for more accurate and stable assignments. Experimental results on benchmark datasets demonstrate 10 to 20 percent higher NMI and ARI scores compared to Louvain, Newman-Girvan, LPA, Fast-Greedy, and a recent GNN-based competitor (MAGI, KDD 2024). Additional evaluations with modularity and F1-scores confirm the superiority of ALCMeans. Ablation studies highlight the critical contributions of each component. Despite its reliance on DeepWalk parameters and increased runtime relative to lightweight heuristics, ALCMeans consistently outperforms state-of-the-art methods. This makes it a promising tool for real-world network analysis.

Selecting a clustering algorithm and its hyperparameters without labels is a common difficulty in engineering machine learning pipelines that work with unsupervised analysis of sensor, image, or process data. Clustering validation indices (CVIs) provide internal scores for ranking candidate clusterings, but most popular CVIs are built from Euclidean compactness and separation terms and so tend to favour compact, convex partitions. Their performance is known to degrade on non convex, irregular, or variable density data, where kernel transformations or alternative distance measures are typically used at the cost of additional tuning and computation. This paper introduces the Central Description Length (CDL) clustering validation index. CDL uses the observed within cluster compactness, the estimated cluster centers, and the estimated cluster covariances to compute a probabilistic upper bound on the description length associated with the unobservable true cluster centers. The bound condenses intra cluster compactness and centroid displacement into a single computable quantity and is evaluated on the partition produced by any clustering algorithm. The implementation uses only observable quantities (the data, the partition, the estimated centers, and the estimated covariances) and does not use ground truth labels. On synthetic benchmarks with non convex and arbitrary shape clusters, CDL-CVI selected the reference number of clusters more often and reached higher Adjusted Rand Index (ARI) values than the conventional CVIs we tested, without an additional kernel preprocessing stage. On image benchmarks (MNIST, CIFAR-10, STL-10) clustered from frozen unsupervised embeddings, CDL-CVI returned cluster numbers close to the reference class counts across K-means, DBSCAN, and spectral clustering in the reported trials.

This paper introduces an algorithm-agnostic approach to feature-based time series clustering via amortized neural inference. By training neural networks to approximate the optimal partitioning rule from simulated data, the proposed framework reduces reliance on conventional clustering methods, such as $K$-means, $K$-medoids, or hierarchical clustering, and their associated objective functions and heuristics. Leveraging statistical features, such as autocorrelations and quantile autocorrelations, the approach learns a data-driven affinity structure from which clustering partitions can be recovered, without requiring explicit prior specification of cluster shapes or structures. In addition, one version of the method can automatically determine the number of clusters, avoiding ad-hoc selection procedures. Comprehensive empirical studies show that the proposed framework achieves competitive or superior clustering accuracy relative to traditional methods, even in challenging scenarios where competing techniques are provided with the true number of clusters. An application to financial time series of stock returns illustrates its practical utility. By reducing the need for algorithm selection and calibration, the proposed framework opens new possibilities for automated, adaptive, and data-driven clustering of temporal data across scientific and industrial domains.

Estimating the number of components is a fundamental challenge in unsupervised learning, particularly when dealing with high-dimensional data with many components or severely imbalanced component sizes. This paper addresses this challenge for classical Gaussian mixture models. The proposed estimator is simple: center the data, compute the singular values of the centered matrix, and count those above a threshold. No iterative fitting, no likelihood calculation, and no prior knowledge of the number of components are required. We prove that, under a mild separation condition on the component centers, the estimator consistently recovers the true number of components. The result holds in high-dimensional settings where the dimension can be much larger than the sample size. It also holds when the number of components grows to the smaller of the dimension and the sample size, even under severe imbalance among component sizes. Computationally, the method is extremely fast: for example, it processes ten million samples in one hundred dimensions within one minute. Extensive experimental studies confirm its accuracy in challenging settings such as high dimensionality, many components, and severe class imbalance.

Determining the number of clusters is a central challenge in unsupervised learning, where ground-truth labels are unavailable. The Silhouette coefficient is a widely used internal validation metric for this task, yet its standard micro-averaged form tends to favor larger clusters under size imbalance. Macro-averaging mitigates this bias by weighting clusters equally, but may overemphasize noise from under-represented groups. We introduce Composite Silhouette, an internal criterion for cluster-count selection that aggregates evidence across repeated subsampled clusterings rather than relying on a single partition. For each subsample, micro- and macro-averaged Silhouette scores are combined through an adaptive convex weight determined by their normalized discrepancy and smoothed by a bounded nonlinearity; the final score is then obtained by averaging these subsample-level composites. We establish key properties of the criterion and derive finite-sample concentration guarantees for its subsampling estimate. Experiments on synthetic and real-world datasets show that Composite Silhouette effectively reconciles the strengths of micro- and macro-averaging, yielding more accurate recovery of the ground-truth number of clusters.

One of the main challenges in data mining is choosing the optimal number of clusters without prior information. Notably, existing methods are usually in the philosophy of cluster validation and hence have underlying assumptions on data distribution, which prevents their application to complex data such as large-scale images and high-dimensional data from the real world. In this regard, we propose an approach named CNMBI. Leveraging the distribution information inherent in the data space, we map the target task as a dynamic comparison process between cluster centers regarding positional behavior, without relying on the complete clustering results and designing the complex validity index as before. Bipartite graph theory is then employed to efficiently model this process. Additionally, we find that different samples have different confidence levels and thereby actively remove low-confidence ones, which is, for the first time to our knowledge, considered in cluster number determination. CNMBI is robust and allows for more flexibility in the dimension and shape of the target data (e.g., CIFAR-10 and STL-10). Extensive comparison studies with state-of-the-art competitors on various challenging datasets demonstrate the superiority of our method.

Federated Clustering (FC) is an emerging and promising solution in exploring data distribution patterns from distributed and privacy-protected data in an unsupervised manner. Existing FC methods implicitly rely on the assumption that clients are with a known number of uniformly sized clusters. However, the true number of clusters is typically unknown, and cluster sizes are naturally imbalanced in real scenarios. Furthermore, the privacy-preserving transmission constraints in federated learning inevitably reduce usable information, making the development of robust and accurate FC extremely challenging. Accordingly, we propose a novel FC framework named Fed-$k^*$-HC, which can automatically determine an optimal number of clusters $k^*$ based on the data distribution explored through hierarchical clustering. To obtain the global data distribution for $k^*$ determination, we let each client generate micro-subclusters. Their prototypes are then uploaded to the server for hierarchical merging. The density-based merging design allows exploring clusters of varying sizes and shapes, and the progressive merging process can self-terminate according to the neighboring relationships among the prototypes to determine $k^*$. Extensive experiments on diverse datasets demonstrate the FC capability of the proposed Fed-$k^*$-HC in accurately exploring a proper number of clusters.

Selecting the number of clusters remains a fundamental challenge in unsupervised learning. Existing approaches typically focus on identifying a single "optimal" partition, often overlooking statistically meaningful structure present across multiple resolutions. We introduce ElbowSig, a general inferential framework for assessing clustering structure over a range of resolutions. The method formalizes the elbow heuristic by defining a normalized discrete curvature statistic based on the sequence of within-cluster heterogeneity values, and evaluates its significance relative to a null distribution of unstructured data. This yields hypothesis tests across resolutions, enabling simultaneous inference at multiple clustering scales. We derive the asymptotic behavior of the null statistic in both large-sample and high-dimensional regimes, characterizing its limiting form and variability. Because it depends only on the heterogeneity sequence, ElbowSig is compatible with a wide range of clustering algorithms, including hard, fuzzy, and model-based methods. Experiments on synthetic and real datasets show that the procedure controls Type-I error under unstructured data while providing power to detect multiscale organization, revealing structure that is often missed by single-resolution selection criteria.

Prior-data Fitted Networks (PFNs) have reframed supervised tabular learning as single-pass in-context inference without per-dataset optimization. Extending this paradigm to clustering is appealing yet fundamentally more challenging, due to absent supervision, unknown cluster cardinality, and label switching inherent to partition outputs. Existing PFN-based clustering methods address these challenges only partially, either requiring known cardinality as input or relying on unstable label-ordering conventions and overly restrictive synthetic priors. We introduce TabClustPFN, a clustering PFN that resolves all challenges jointly through co-designed prior, objective, and architecture. Our hybrid pretraining prior captures heterogeneous real-tabular geometry; our decoupled partition inference network--cardinality inference network architecture jointly infers cluster assignments and cardinality in a single pass; and our SoftARI training objective is permutation-invariant by construction, eliminating the need for any label-ordering convention. On a 44 curated real-world tabular benchmark, TabClustPFN achieves state-of-the-art clustering performance against classical, deep, and amortized baselines, with runtime comparable to efficient baselines. Code is available at https://github.com/Tianqi-Zhao/TabClustPFN.