An optimal transport approach for selecting a representative subsample with application in efficient kernel density estimation

TL;DR

This paper introduces a novel model-free subsampling method based on optimal transport, which efficiently selects representative samples for accurate kernel density estimation, overcoming computational and theoretical limitations of existing methods.

Contribution

The paper proposes an optimal transport-based subsampling approach with an adaptive algorithm and theoretical guarantees for density estimation convergence.

Findings

Superior performance on synthetic datasets

Effective in real-world data applications

Theoretical convergence rate established

Abstract

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.