We study data clustering problems with \ell_p -norm objectives (e.g. The dataset consists of n points, and we want to find k centers such that (a) the objective is minimized, while (b) respecting the individual fairness constraint that every point v has a center within a distance at most r(v), where r(v) is v's distance to its (n/k) th nearest point. Jung, Kannan, and Lutz [FORC 2020] introduced this concept and designed a clustering algorithm with provable (approximate) fairness and objective guarantees for the \ell_\infty or \textsc{ k -Center} objective. Empirically, their algorithms outperform Jung et. In this paper, our main contribution is to use Linear Programming (LP) techniques to obtain better algorithms for this problem, both in theory and in practice.

We consider the widely-used average-linkage, single-linkage, and Ward's methods for computing hierarchical clusterings of high-dimensional Euclidean inputs. It is easy to show that there is no efficient implementation of these algorithms in high dimensional Euclidean space since it implicitly requires to solve the closest pair problem, a notoriously difficult problem. However, how fast can these algorithms be implemented if we allow approximation? More precisely: these algorithms successively merge the clusters that are at closest average (for average-linkage), minimum distance (for single-linkage), or inducing the least sum-of-square error (for Ward's). We ask whether one could obtain a significant running-time improvement if the algorithm can merge \gamma -approximate closest clusters (namely, clusters that are at distance (average, minimum, or sum-of-square error) at most \gamma times the distance of the closest clusters).

Spectral clustering and k -means, both as two major traditional clustering methods, are still attracting a lot of attention, although a variety of novel clustering algorithms have been proposed in recent years. Firstly, a unified framework of k -means and ratio-cut is revisited, and a novel and efficient clustering algorithm is then proposed based on this framework. The time and space complexity of our method are both linear with respect to the number of samples, and are independent of the number of clusters to construct, more importantly. These properties mean that it is easily scalable and applicable to large practical problems. Extensive experiments on 12 real-world benchmark and 8 facial datasets validate the advantages of the proposed algorithms compared to the state-of-the-art clustering algorithms.

In the Taiwanese judicial system, Cause of Actions (COAs) are essential for identifying relevant legal judgments. However, the lack of standardized COA labeling creates challenges in filtering cases using basic methods. This research addresses this issue by leveraging embedding and clustering techniques to analyze the similarity between COAs based on cited legal articles. The study implements various similarity measures, including Dice coefficient and Pearson's correlation coefficient. An ensemble model combines rankings, and social network analysis identifies clusters of related COAs. This approach enhances legal analysis by revealing inconspicuous connections between COAs, offering potential applications in legal research beyond civil law.

Sparse Mixture-of-Experts (SMoE) models represent a significant breakthrough in large language model development. These models enable performance improvements without a proportional increase in inference costs. By selectively activating a small set of parameters during task execution, SMoEs enhance model capacity. However, their deployment remains challenging due to the substantial memory footprint required to accommodate the growing number of experts. To address this challenge, we propose Hierarchical Clustering for Sparsely activated Mixture of Experts (HC-SMoE), a task-agnostic expert merging framework that reduces SMoE model parameters without retraining. Unlike previous methods, HC-SMoE employs hierarchical clustering based on expert outputs. This approach ensures that the merging process remains unaffected by routing decisions. We validate our approach through extensive experiments on eight zero-shot language tasks and demonstrate its effectiveness in large-scale SMoE models such as Qwen and Mixtral. Our comprehensive results demonstrate that HC-SMoE consistently achieves strong performance, which highlights its potential for real-world deployment. The exponential growth in model parameters for Transformer-based architectures in natural language processing (NLP) has led to significant performance improvements across various tasks (Chowdhery et al., 2022; OpenAI et al., 2024; Team et al., 2024). Nevertheless, this increase in size has resulted in challenges for real-world deployment and accessibility due to heightened inference latency and computational requirements (Bommasani et al., 2022) Sparsely activated Mixture of Experts (SMoE) models have emerged as a promising solution to this challenge.

Clustering is an important exploratory data analysis technique to group objects based on their similarity. The widely used K -means clustering method relies on some notion of distance to partition data into a fewer number of groups. In the Euclidean space, centroid-based and distance-based formulations of the K -means are equivalent. In modern machine learning applications, data often arise as probability distributions and a natural generalization to handle measure-valued data is to use the optimal transport metric. Due to non-negative Alexandrov curvature of the Wasserstein space, barycenters suffer from regularity and non-robustness issues.

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.

We address the classical problem of hierarchical clustering, but in a framework where one does not have access to a representation of the objects or their pairwise similarities. Instead, we assume that only a set of comparisons between objects is available, that is, statements of the form objects i and j are more similar than objects k and l.'' Such a scenario is commonly encountered in crowdsourcing applications. The focus of this work is to develop comparison-based hierarchical clustering algorithms that do not rely on the principles of ordinal embedding. We show that single and complete linkage are inherently comparison-based and we develop variants of average linkage.

Clustering is a ubiquitous task in data science. Compared to the commonly used k-means clustering, k-medoids clustering requires the cluster centers to be actual data points and supports arbitrary distance metrics, which permits greater interpretability and the clustering of structured objects. Current state-of-the-art k-medoids clustering algorithms, such as Partitioning Around Medoids (PAM), are iterative and are quadratic in the dataset size n for each iteration, being prohibitively expensive for large datasets. We propose BanditPAM, a randomized algorithm inspired by techniques from multi-armed bandits, that reduces the complexity of each PAM iteration from O(n 2) to O(nlogn) and returns the same results with high probability, under assumptions on the data that often hold in practice. As such, BanditPAM matches state-of-the-art clustering loss while reaching solutions much faster.

Clustering, a fundamental activity in unsupervised learning, is notoriously difficult when the feature space is high-dimensional. Fortunately, in many realistic scenarios, only a handful of features are relevant in distinguishing clusters. This has motivated the development of sparse clustering techniques that typically rely on k-means within outer algorithms of high computational complexity. Current techniques also require careful tuning of shrinkage parameters, further limiting their scalability. In this paper, we propose a novel framework for sparse k-means clustering that is intuitive, simple to implement, and competitive with state-of-the-art algorithms.