  Subsampling methods aim to select a subsample as a surrogate for the observed
sample. Such methods have been used pervasively in large-scale data analytics,
active learning, and privacy-preserving analysis in recent decades. Instead of
model-based methods, in this paper, we study model-free subsampling methods,
which aim to identify a subsample that is not confined by model assumptions.
Existing model-free subsampling methods are usually built upon clustering
techniques or kernel tricks. Most of these methods suffer from either a large
computational burden or a theoretical weakness. In particular, the theoretical
weakness is that the empirical distribution of the selected subsample may not
necessarily converge to the population distribution. Such computational and
theoretical limitations hinder the broad applicability of model-free
subsampling methods in practice. We propose a novel model-free subsampling
method by utilizing optimal transport techniques. Moreover, we develop an
efficient subsampling algorithm that is adaptive to the unknown probability
density function. Theoretically, we show the selected subsample can be used for
efficient density estimation by deriving the convergence rate for the proposed
subsample kernel density estimator. We also provide the optimal bandwidth for
the proposed estimator. Numerical studies on synthetic and real-world datasets
demonstrate the performance of the proposed method is superior.
