Greedy MMD Without Replacement: Selecting and Stopping for Mixture Separation

This paper studies how to split an unlabelled sample drawn from a mixture $\theta P+(1-\theta )Q$ when a separate reference sample from $P$ is available. Greedy maximum mean discrepancy (MMD) selection without replacement chooses distinct observations to approximate $P$, leaving a remainder intended to approximate $Q$. Success depends on order and stopping: early choices may remove observations needed later, while a wrong stopping size can leave a poor remainder even when the selected observations fit $P$. An error bound tracks the effect of earlier selections. Under two geometric conditions and appropriate stopping, convergence rates are established for the MMD errors of the selected group relative to $P$ and the remainder relative to $Q$. Simulations and real-data experiments illustrate the effect of stopping and support the predicted convergence behaviour.