Image Clustering

Functional data present unique challenges for clustering due to their infinite-dimensional nature and potential sensitivity to outliers. An extension of the OCLUST algorithm to the functional setting is proposed to address these issues. The approach leverages the OCLUST framework, creating a robust method to cluster curves and trim outliers. The methodology is evaluated on both simulated and real-world functional datasets, demonstrating strong performance in clustering and outlier identification.

Unsupervised feature selection is critical for improving clustering performance in high-dimensional data, where irrelevant features can obscure meaningful structure. In this work, we propose the Minkowski weighted $k$-means++, a novel initialisation strategy for the Minkowski Weighted $k$-means. Our initialisation selects centroids probabilistically using feature relevance estimates derived from the data itself. Building on this, we propose two new feature selection algorithms, FS-MWK++, which aggregates feature weights across a range of Minkowski exponents identifying stable and informative features, and SFS-MWK++, a scalable variant based on subsampling. We support our approach with a theoretical analysis, demonstrating that, under explicit assumptions on noise features and cluster structure, relevant features are assigned consistently higher weights than noise features across a range of Minkowski exponents. Our software can be found at https://github.com/xzhang4-ops1/FSMWK.

Recovering the underlying $k$-clustering of a set $U$ of $n$ points by asking pair-wise same-cluster queries has garnered significant interest in the past few years. Given a query $S \subset U$, $|S|=2$, the oracle returns "yes" if the points are in the same cluster and "no" otherwise. For adaptive algorithms, the query complexity is known to be $Θ(nk)$, while non-adaptive algorithms are extremely limited: even for $k=3$, such algorithms require $Ω(n^2)$ queries, matching the trivial upper bound. However, non-adaptivity is highly desirable since it allows queries to be asked in parallel. To break the quadratic barrier for non-adaptive queries, we study a natural generalization of this problem to subset queries for $|S|>2$, where the oracle returns the number of clusters intersecting $S$. Previous work obtained an $O(n)$ query adaptive algorithm, but the realm of non-adaptive algorithms remained completely unknown. In this paper, we give the first non-adaptive algorithms for clustering with subset queries. Our main result is a non-adaptive algorithm making $O(n \log k \cdot (\log k + \log\log n)^2)$ queries, improving to $O(n \log \log n)$ when $k$ is constant. In addition to non-adaptivity, we make other practical considerations, such as enforcing a bound, $s$, on the query size. We show $Ω(\max(n^2/s^2,n))$ queries are necessary and obtain algorithms making $\smash{\widetilde{O}(n^2k/s^2)}$ queries for any $s \leq \sqrt{n}$ and $\smash{\widetilde{O}(n^2/s)}$ queries for any $s \leq n$. Finally, we obtain improved upper bounds when the clusters are roughly balanced, and when the algorithm is allowed two rounds of adaptivity.

In recent years, much of the research on clustering algorithms has primarily focused on enhancing their accuracy and efficiency, frequently at the expense of interpretability. However, as these methods are increasingly being applied in high-stakes domains such as healthcare, finance, and autonomous systems, the need of transparent and interpretable clustering outcomes has become a critical concern. This is not only necessary for gaining user trust but also for satisfying the growing ethical and regulatory demands in these fields. Ensuring that decisions derived from clustering algorithms can be clearly understood and justified is now a fundamental requirement. To address this need, this paper provides a comprehensive and structured review of the current state of explainable clustering algorithms, identifying key criteria to distinguish between various methods. These insights can effectively assist researchers in making informed decisions about the most suitable explainable clustering methods for specific application contexts, while also promoting the development and adoption of clustering algorithms that are both efficient and transparent. For convenient access and reference, an open repository organizes representative and emerging interpretable clustering methods under the taxonomy proposed in this survey, available at https://hulianyu.xyz/Interpretable-Clustering-Repository

Clustering problems such as $k$-means and $k$-median are staples of unsupervised learning, and many algorithmic techniques have been developed to tackle their numerous aspects. In this paper, we focus on the class of greedy approximation algorithm, that attracted less attention than local-search or primal-dual counterparts. In particular, we study the recursive greedy algorithm developed by Mettu and Plaxton [SIAM J. Comp 2003]. We provide a simplification of the algorithm, allowing for faster implementation, in graph metrics or in Euclidean space, where our algorithm matches or improves the state-of-the-art.

Graph clustering, an important unsupervised problem, has been shown to be more resistant to advances in Graph Neural Networks (GNNs). In addition, almost all clustering methods focus on homophilic graphs and ignore heterophily. This significantly limits their applicability in practice, since real-world graphs exhibit a structural disparity and cannot simply be classified as homophily and heterophily. Thus, a principled way to handle practical graphs is urgently needed. To fill this gap, we provide a novel solution with theoretical support. Interestingly, we find that most homophilic and heterophilic edges can be correctly identified on the basis of neighbor information. Motivated by this finding, we construct two graphs that are highly homophilic and heterophilic, respectively. They are used to build low-pass and high-pass filters to capture holistic information. Important features are further enhanced by the squeeze-and-excitation block. We validate our approach through extensive experiments on both homophilic and heterophilic graphs. Empirical results demonstrate the superiority of our method compared to state-of-the-art clustering methods.

In this paper, we consider the problem of partitioning a small data sample of size $n$ drawn from a mixture of $2$ sub-gaussian distributions in $\mathbb{R}^p$. We consider semidefinite programming relaxations of an integer quadratic program that is formulated essentially as finding the maximum cut on a graph, where edge weights in the cut represent dissimilarity scores between two nodes based on their $p$ features. We define the signal-to-noise ratio (SNR) as $s^2 := \min\{n p γ^2, Δ^2\}$, where $Δ^2 := p γ$ denotes the $\ell_2^2$ distance between the two cluster centers. Our contributions are twofold. First, we provide a unified framework for analyzing three computationally efficient algorithms: SDP1, BalancedSDP, and Spectral clustering, yielding universal polynomial-rate misclassification guarantees for all three algorithms. Moreover, our theory allows for partial recovery (success rate $< 100\%$) as long as $s^2$ is lower bounded by a constant. Second, we prove that the misclassification errors for SDP1 and BalancedSDP decay exponentially with respect to the SNR $s^2$ and the BalancedSDP requires no explicit debiasing when the two clusters have equal sizes. To our knowledge, this is the first time such results are obtained for semidefinite relaxations of MAX CUT in population clustering. We provide simulation evidence illuminating the theoretical predictions.

Matrix-variate distributions are a relatively recent addition to the model-based clustering literature, thereby making it possible to analyze data in matrix form with complex structure such as images and time series. Due to its recent appearance, there is limited literature on matrix-variate data, with even less on dealing with outliers in these models. An approach for clustering matrix-variate normal data with outliers is discussed. The approach, which uses the distribution of subset log-likelihoods, extends the OCLUST algorithm to matrix-variate normal data and uses an iterative approach to detect and trim outliers.

In recent years, there has been a surge in effort to formalize notions of fairness in machine learning. We focus on centroid clustering--one of the fundamental tasks in unsupervised machine learning. We propose a new axiom ``proportionally representative fairness'' (PRF) that is designed for clustering problems where the selection of centroids reflects the distribution of data points and how tightly they are clustered together. Our fairness concept is not satisfied by existing fair clustering algorithms. We design efficient algorithms to achieve PRF both for unconstrained and discrete clustering problems. Our algorithm for the unconstrained setting is also the first known polynomial-time approximation algorithm for the well-studied Proportional Fairness (PF) axiom. Our algorithm for the discrete setting also matches the best known approximation factor for PF.

Vertex-level clustering for directed graphs (digraphs) remains challenging as edge directionality breaks the key assumptions underlying popular spectral methods, which also incur the overhead of eigen-decomposition. This paper proposes Parametrized Power-Iteration Clustering (ParPIC), a random-walk-based clustering method for weakly connected digraphs. This builds over the Power-Iteration Clustering paradigm, which uses the rows of the iterated diffusion operator as a data embedding. ParPIC has three important features: the use of parametrized reversible random walk operators, the automatic tuning of the diffusion time, and the efficient truncation of the final embedding, which produces low-dimensional data representations and reduces complexity. Empirical results on synthetic and real-world graphs demonstrate that ParPIC achieves competitive clustering accuracy with improved scalability relative to spectral and teleportation-based methods.

K-Means clustering algorithm is one of the most commonly used clustering algorithms because of its simplicity and efficiency. K-Means clustering algorithm based on Euclidean distance only pays attention to the linear distance between Euclidean distance is an efficient and interpretable similarity measurement, but its effectiveness may deteriorate in sample spaces with anisotropic structures, redundant features, or complex feature interactions. In this paper, we propose a novel distance metric called View distance. Inspired by orthographic projection, the proposed metric projects the sample space onto $n(n-1)/2$ two-dimensional planes and defines the final distance as the sum of the Euclidean distances across the projected planes. Theoretical derivations verify that the View distance strictly satisfies the metric axioms and norm constraints. Beyond that, the View distance achieves feature coupling through projection and not only enables constant features to indirectly participate in distance calculation, but also suppresses interference from redundant features while exhibiting anisotropic geometric properties. Furthermore, to address the high computational complexity and poor scalability of full-projection View distance, we propose a two-dimensional projection plane selection strategy based on iterative Maximum Weight Matching, which reduces the computational complexity of distance calculation from $\mathcal{O}(n^2)$ to $\mathcal{O}(k)$. Extensive experiments on 12 diverse datasets demonstrate that View distance and the deterministic selection strategy provide competitive or superior performance compared with Euclidean distance and other $L_{p}$ metrics while maintaining strong interpretability and computational efficiency. The View distance provides a new perspective and option for similarity measurements.

Building trustworthy machine learning algorithms for biological applications requires adapting to data heterogeneity from different sources, batches, distributions, or studies. We propose the 'Cross-Cluster Weighted Forest' (CCWF), an ensembling approach that explicitly leverages heterogeneity in the feature distribution to produce more accurate and more generalizable predictors than the standard Random Forest in cases when data can be naturally clustered. CCWF generalizes the RF architecture to an outer unsupervised layer, supervised subtasks, and ensembling. Specifically it involves unsupervised clustering of the training data, fitting a Random Forest on each cluster, and combining the forests via stacked regression weights that reward cross-cluster generalizability. We provide a theoretical analysis of an analytically tractable forest model showing that cluster-based ensembling is asymptotically more accurate than training a single forest on the full data, with the gain driven by bias reduction. In simulations, we find that CCWF is robust across data-generating regimes and outcome models; furthermore, we explore the influence of data partitioning and ensemble weighting strategies on the benefits of our method. Finally, we apply our approach to cancer molecular profiling and gene expression datasets that are naturally divisible into clusters; in both simulations and real data examples, we illustrate that our approach outperforms classic Random Forest by margins of 30-40%, aligning with our theoretical results. Overall, we show that CCWF provides a statistically grounded prediction algorithm for data spanning multiple domains or sub-populations, a structure common in biological applications.