Vector Quantization (VQ) is a widely used technique in machine learning and data compression, valued for its simplicity and interpretability. Among hard VQ methods, k-medoids clustering and Kernel Density Estimation (KDE) approaches represent two prominent yet seemingly unrelated paradigms -- one distance-based, the other rooted in probability density matching. In this paper, we investigate their connection through the lens of Quadratic Unconstrained Binary Optimization (QUBO). We compare a heuristic QUBO formulation for k-medoids, which balances centrality and diversity, with a principled QUBO derived from minimizing Maximum Mean Discrepancy in KDE-based VQ. Surprisingly, we show that the KDE-QUBO is a special case of the k-medoids-QUBO under mild assumptions on the kernel's feature map. This reveals a deeper structural relationship between these two approaches and provides new insight into the geometric interpretation of the weighting parameters used in QUBO formulations for VQ.
@article{arxiv.2506.04786,
title = {Kernel $k$-Medoids as General Vector Quantization},
author = {Thore Gerlach and Sascha Mücke and Christian Bauckhage},
journal= {arXiv preprint arXiv:2506.04786},
year = {2025}
}