Fair clustering has traditionally focused on ensuring equitable group representation or equalizing group-specific clustering costs. However, Dickerson et al. (2025) recently showed that these fairness notions may yield undesirable or unintuitive clustering outcomes and advocated for a welfare-centric clustering approach that models the utilities of the groups. In this work, we model group utilities based on both distances and proportional representation and formalize two optimization objectives based on welfare-centric clustering: the Rawlsian (Egalitarian) objective and the Utilitarian objective. We introduce novel algorithms for both objectives and prove theoretical guarantees for them. Empirical evaluations on multiple real-world datasets demonstrate that our methods significantly outperform existing fair clustering baselines.

The proliferation of Large Language Models (LLMs) is challenged by hallucinations, critical failure modes where models generate non-factual, nonsensical or unfaithful text. This paper introduces Semantic Divergence Metrics (SDM), a novel lightweight framework for detecting Faithfulness Hallucinations -- events of severe deviations of LLMs responses from input contexts. We focus on a specific implementation of these LLM errors, {confabulations, defined as responses that are arbitrary and semantically misaligned with the user's query. Existing methods like Semantic Entropy test for arbitrariness by measuring the diversity of answers to a single, fixed prompt. Our SDM framework improves upon this by being more prompt-aware: we test for a deeper form of arbitrariness by measuring response consistency not only across multiple answers but also across multiple, semantically-equivalent paraphrases of the original prompt. Methodologically, our approach uses joint clustering on sentence embeddings to create a shared topic space for prompts and answers. A heatmap of topic co-occurances between prompts and responses can be viewed as a quantified two-dimensional visualization of the user-machine dialogue. We then compute a suite of information-theoretic metrics to measure the semantic divergence between prompts and responses. Our practical score, $\mathcal{S}_H$, combines the Jensen-Shannon divergence and Wasserstein distance to quantify this divergence, with a high score indicating a Faithfulness hallucination. Furthermore, we identify the KL divergence KL(Answer $||$ Prompt) as a powerful indicator of \textbf{Semantic Exploration}, a key signal for distinguishing different generative behaviors. These metrics are further combined into the Semantic Box, a diagnostic framework for classifying LLM response types, including the dangerous, confident confabulation.

Most work on estimating mixture models assumes an iid sampling scheme.

Spectral clustering is popular among practitioners and theoreticians alike.

Thus, we restrict our search for a constrained clustering approach to the class of deep generative models. Although these models have been successfully used in the unsupervised setting (Jiang et al., 2017; Dilokthanakul et al., 2016), their application to constrained clustering has been under-explored.

Clustering is an important exploratory data analysis technique to group objects based on their similarity.

In most real world decision problems, the decision maker (DM) faces uncertainty either in the objective function that he aims to optimize, or some of the constraints that he needs to satisfy.

Accurate cell type annotation is a crucial step in analyzing single-cell RNA sequencing (scRNA-seq) data, which provides valuable insights into cellular heterogeneity. However, due to the high dimensionality and prevalence of zero elements in scRNA-seq data, traditional clustering methods face significant statistical and computational challenges. While some advanced methods use graph neural networks to model cell-cell relationships, they often depend on static graph structures that are sensitive to noise and fail to capture the long-tailed distribution inherent in single-cell populations.To address these limitations, we propose scAGC, a single-cell clustering method that learns adaptive cell graphs with contrastive guidance. Our approach optimizes feature representations and cell graphs simultaneously in an end-to-end manner. Specifically, we introduce a topology-adaptive graph autoencoder that leverages a differentiable Gumbel-Softmax sampling strategy to dynamically refine the graph structure during training. This adaptive mechanism mitigates the problem of a long-tailed degree distribution by promoting a more balanced neighborhood structure. To model the discrete, over-dispersed, and zero-inflated nature of scRNA-seq data, we integrate a Zero-Inflated Negative Binomial (ZINB) loss for robust feature reconstruction. Furthermore, a contrastive learning objective is incorporated to regularize the graph learning process and prevent abrupt changes in the graph topology, ensuring stability and enhancing convergence. Comprehensive experiments on 9 real scRNA-seq datasets demonstrate that scAGC consistently outperforms other state-of-the-art methods, yielding the best NMI and ARI scores on 9 and 7 datasets, respectively.Our code is available at Anonymous Github.