With grid operators confronting rising uncertainty from renewable integration and a broader push toward electrification, Demand-Side Management (DSM) -- particularly Demand Response (DR) -- has attracted significant attention as a cost-effective mechanism for balancing modern electricity systems. Unprecedented volumes of consumption data from a continuing global deployment of smart meters enable consumer segmentation based on real usage behaviours, promising to inform the design of more effective DSM and DR programs. However, existing clustering-based segmentation methods insufficiently reflect the behavioural diversity of consumers, often relying on rigid temporal alignment, and faltering in the presence of anomalies, missing data, or large-scale deployments. To address these challenges, we propose a novel two-stage clustering framework -- Clustered Representations Optimising Consumer Segmentation (CROCS). In the first stage, each consumer's daily load profiles are clustered independently to form a Representative Load Set (RLS), providing a compact summary of their typical diurnal consumption behaviours. In the second stage, consumers are clustered using the Weighted Sum of Minimum Distances (WSMD), a novel set-to-set measure that compares RLSs by accounting for both the prevalence and similarity of those behaviours. Finally, community detection on the WSMD-induced graph reveals higher-order prototypes that embody the shared diurnal behaviours defining consumer groups, enhancing the interpretability of the resulting clusters. Extensive experiments on both synthetic and real Australian smart meter datasets demonstrate that CROCS captures intra-consumer variability, uncovers both synchronous and asynchronous behavioural similarities, and remains robust to anomalies and missing data, while scaling efficiently through natural parallelisation. These results...

We study the Bandit Clustering (BC) problem under the fixed confidence setting, where the objective is to group a collection of data sequences (arms) into clusters through sequential sampling from adaptively selected arms at each time step while ensuring a fixed error probability at the stopping time. We consider a setting where arms in a cluster may have different distributions. Unlike existing results in this setting, which assume Gaussian-distributed arms, we study a broader class of vector-parametric distributions that satisfy mild regularity conditions. Existing asymptotically optimal BC algorithms require solving an optimization problem as part of their sampling rule at each step, which is computationally costly. We propose an Efficient Bandit Clustering algorithm (EBC), which, instead of solving the full optimization problem, takes a single step toward the optimal value at each time step, making it computationally efficient while remaining asymptotically optimal. We also propose a heuristic variant of EBC, called EBC-H, which further simplifies the sampling rule, with arm selection based on quantities computed as part of the stopping rule. We highlight the computational efficiency of EBC and EBC-H by comparing their per-sample run time with that of existing algorithms. The asymptotic optimality of EBC is supported through simulations on the synthetic datasets. Through simulations on both synthetic and real-world datasets, we show the performance gain of EBC and EBC-H over existing approaches.

Although extensive research exists in spatial modeling, few studies have addressed finite mixture model-based clustering methods for spatial data. Finite mixture models, especially Gaussian mixture models, particularly suffer from high dimensionality due to the number of free covariance parameters. This study introduces a spatial covariance constraint for Gaussian mixture models that requires only four free parameters for each component, independent of dimensionality. Using a coordinate system, the spatially constrained Gaussian mixture model enables clustering of multi-way spatial data and inference of spatial patterns. The parameter estimation is conducted by combining the expectation-maximization (EM) algorithm with the generalized least squares (GLS) estimator. Simulation studies and applications to Raman spectroscopy data are provided to demonstrate the proposed model.

Clustering with bandit feedback refers to the problem of partitioning a set of items, where the clustering algorithm can sequentially query the items to receive noisy observations. The problem is formally posed as the task of partitioning the arms of an N-armed stochastic bandit according to their underlying distributions, grouping two arms together if and only if they share the same distribution, using samples collected sequentially and adaptively. This setting has gained attention in recent years due to its applicability in recommendation systems and crowdsourcing. Existing works on clustering with bandit feedback rely on a strong assumption that the underlying distributions are sub-Gaussian. As a consequence, the existing methods mainly cover settings with linearly-separable clusters, which has little practical relevance. We introduce a framework of ``nonparametric clustering with bandit feedback'', where the underlying arm distributions are not constrained to any parametric, and hence, it is applicable for active clustering of real-world datasets. We adopt a kernel-based approach, which allows us to reformulate the nonparametric problem as the task of clustering the arms according to their kernel mean embeddings in a reproducing kernel Hilbert space (RKHS). Building on this formulation, we introduce the KABC algorithm with theoretical correctness guarantees and analyze its sampling budget. We introduce a notion of signal-to-noise ratio for this problem that depends on the maximum mean discrepancy (MMD) between the arm distributions and on their variance in the RKHS. Our algorithm is adaptive to this unknown quantity: it does not require it as an input yet achieves instance-dependent guarantees.

This paper, introducing a novel method in philo-matics, draws on Wittgenstein's concept of family resemblance from analytic philosophy to develop a clustering algorithm for machine learning. According to Wittgenstein's Philosophical Investigations (1953), family resemblance holds that members of a concept or category are connected by overlapping similarities rather than a single defining property. Consequently, a family of entities forms a chain of items sharing overlapping traits. This philosophical idea naturally lends itself to a graph-based approach in machine learning. Accordingly, we propose the Wittgenstein's Family Resemblance (WFR) clustering algorithm and its kernel variant, kernel WFR. This algorithm computes resemblance scores between neighboring data instances, and after thresholding these scores, a resemblance graph is constructed. The connected components of this graph define the resulting clusters. Simulations on benchmark datasets demonstrate that WFR is an effective nonlinear clustering algorithm that does not require prior knowledge of the number of clusters or assumptions about their shapes.

Topological methods have the potential of exploring data clouds without making assumptions on their the structure. Here we propose a hierarchical topological clustering algorithm that can be implemented with any distance choice. The persistence of outliers and clusters of arbitrary shape is inferred from the resulting hierarchy. We demonstrate the potential of the algorithm on selected datasets in which outliers play relevant roles, consisting of images, medical and economic data. These methods can provide meaningful clusters in situations in which other techniques fail to do so.

Hierarchical clustering methods offer an intuitive and powerful way to model a wide variety of data sets. However, the assumption of a fixed hierarchy is often overly restrictive when working with data generated over a period of time: We expect both the structure of our hierarchy, and the parameters of the clusters, to evolve with time. In this paper, we present a distribution over collections of time-dependent, infinite-dimensional trees that can be used to model evolving hierarchies, and present an efficient and scalable algorithm for performing approximate inference in such a model. We demonstrate the efficacy of our model and inference algorithm on both synthetic data and real-world document corpora.

We propose Tree Variational Autoencoder (TreeVAE), a new generative hierarchical clustering model that learns a flexible tree-based posterior distribution over latent variables. TreeVAE hierarchically divides samples according to their intrinsic characteristics, shedding light on hidden structures in the data. It adapts its architecture to discover the optimal tree for encoding dependencies between latent variables. The proposed tree-based generative architecture enables lightweight conditional inference and improves generative performance by utilizing specialized leaf decoders. We show that TreeVAE uncovers underlying clusters in the data and finds meaningful hierarchical relations between the different groups on a variety of datasets, including real-world imaging data. We present empirically that TreeVAE provides a more competitive log-likelihood lower bound than the sequential counterparts. Finally, due to its generative nature, TreeVAE is able to generate new samples from the discovered clusters via conditional sampling.

We design replicable algorithms in the context of statistical clustering under the recently introduced notion of replicability from Impagliazzo et al. [2022]. According to this definition, a clustering algorithm is replicable if, with high probability, its output induces the exact same partition of the sample space after two executions on different inputs drawn from the same distribution, when its internal randomness is shared across the executions. We propose such algorithms for the statistical $k$-medians, statistical $k$-means, and statistical $k$-centers problems by utilizing approximation routines for their combinatorial counterparts in a black-box manner. In particular, we demonstrate a replicable $O(1)$-approximation algorithm for statistical Euclidean $k$-medians ($k$-means) with $\operatorname{poly}(d)$ sample complexity. We also describe an $O(1)$-approximation algorithm with an additional $O(1)$-additive error for statistical Euclidean $k$-centers, albeit with $\exp(d)$ sample complexity. In addition, we provide experiments on synthetic distributions in 2D using the $k$-means++ implementation from sklearn as a black-box that validate our theoretical results.

We study the problem of fitting an ultrametric distance to a dissimilarity graph in the context of hierarchical cluster analysis. Standard hierarchical clustering methods are specified procedurally, rather than in terms of the cost function to be optimized. We aim to overcome this limitation by presenting a general optimization framework for ultrametric fitting. Our approach consists of modeling the latter as a constrained optimization problem over the continuous space of ultrametrics. So doing, we can leverage the simple, yet effective, idea of replacing the ultrametric constraint with a min-max operation injected directly into the cost function. The proposed reformulation leads to an unconstrained optimization problem that can be efficiently solved by gradient descent methods. The flexibility of our framework allows us to investigate several cost functions, following the classic paradigm of combining a data fidelity term with a regularization. While we provide no theoretical guarantee to find the global optimum, the numerical results obtained over a number of synthetic and real datasets demonstrate the good performance of our approach with respect to state-of-the-art agglomerative algorithms. This makes us believe that the proposed framework sheds new light on the way to design a new generation of hierarchical clustering methods.