We study the deterministic global optimization of the K-Medoids clustering problem. This work proposes a branch and bound (BB) scheme, in which a tailored Lagrangian relaxation method proposed in the 1970s is used to provide a lower bound at each BB node. The lower bounding method already guarantees the maximum gap at the root node. A closed-form solution to the lower bound can be derived analytically without explicitly solving any optimization problems, and its computation can be easily parallelized. Moreover, with this lower bounding method, finite convergence to the global optimal solution can be guaranteed by branching only on the regions of medoids. We also present several tailored bound tightening techniques to reduce the search space and computational cost. Extensive computational studies on 28 machine learning datasets demonstrate that our algorithm can provide a provable global optimal solution with an optimality gap of 0.1% within 4 hours on datasets with up to one million samples. Besides, our algorithm can obtain better or equal objective values than the heuristic method. A theoretical proof of global convergence for our algorithm is also presented.

Clustering is a fundamental task in data science with wide-ranging applications. In k-medoids clustering, cluster centers must be actual datapoints and arbitrary distance metrics may be used; these features allow for greater interpretability of the cluster centers and the clustering of exotic objects in k-medoids clustering, respectively.

Clustering is a fundamental task in data science with wide-ranging applications. In k-medoids clustering, cluster centers must be actual datapoints and arbitrary distance metrics may be used; these features allow for greater interpretability of the cluster centers and the clustering of exotic objects in k-medoids clustering, respectively.

Large language models (LLMs) are increasingly used as agents to solve complex tasks such as question answering (QA), scientific debate, and software development. A standard evaluation procedure aggregates multiple responses from LLM agents into a single final answer, often via majority voting, and compares it against reference answers. However, this process can obscure the quality and distributional characteristics of the original responses. In this paper, we propose a novel evaluation framework based on the empirical cumulative distribution function (ECDF) of cosine similarities between generated responses and reference answers. This enables a more nuanced assessment of response quality beyond exact match metrics. To analyze the response distributions across different agent configurations, we further introduce a clustering method for ECDFs using their distances and the $k$-medoids algorithm. Our experiments on a QA dataset demonstrate that ECDFs can distinguish between agent settings with similar final accuracies but different quality distributions. The clustering analysis also reveals interpretable group structures in the responses, offering insights into the impact of temperature, persona, and question topics.

Clustering is a fundamental task in data science with wide-ranging applications.

Generative models are increasingly used to produce privacy-preserving synthetic data as a safe alternative to sharing sensitive training datasets. However, we demonstrate that such synthetic releases can still leak information about the underlying training samples through structural overlap in the data manifold. We propose a black-box membership inference attack that exploits this vulnerability without requiring access to model internals or real data. The attacker repeatedly queries the generative model to obtain large numbers of synthetic samples, performs unsupervised clustering to identify dense regions of the synthetic distribution, and then analyzes cluster medoids and neighborhoods that correspond to high-density regions in the original training data. These neighborhoods act as proxies for training samples, enabling the adversary to infer membership or reconstruct approximate records. Our experiments across healthcare, finance, and other sensitive domains show that cluster overlap between real and synthetic data leads to measurable membership leakage-even when the generator is trained with differential privacy or other noise mechanisms. The results highlight an under-explored attack surface in synthetic data generation pipelines and call for stronger privacy guarantees that account for distributional neighborhood inference rather than sample-level memorization alone, underscoring its role in privacy-preserving data publishing. Implementation and evaluation code are publicly available at:github.com/Cluster-Medoid-Leakage-Attack.

Clustering is a fundamental task in data science with wide-ranging applications.