Revisit Non-parametric Two-sample Testing as a Semi-supervised Learning Problem

Learning effective data representations is crucial in answering if two samples X and Y are from the same distribution (a.k.a. the non-parametric two-sample testing problem), which can be categorized into: i) learning discriminative representations (DRs) that distinguish between two samples in a supervised-learning paradigm, and ii) learning inherent representations (IRs) focusing on data's inherent features in an unsupervised-learning paradigm. However, both paradigms have issues: learning DRs reduces the data points available for the two-sample testing phase, and learning purely IRs misses discriminative cues. To mitigate both issues, we propose a novel perspective to consider non-parametric two-sample testing as a semi-supervised learning (SSL) problem, introducing the SSL-based Classifier Two-Sample Test (SSL-C2ST) framework. While a straightforward implementation of SSL-C2ST might directly use existing state-of-the-art (SOTA) SSL methods to train a classifier with labeled data (with sample indexes X or Y) and unlabeled data (the remaining ones in the two samples), conventional two-sample testing data often exhibits substantial overlap between samples and violates SSL methods' assumptions, resulting in low test power. Therefore, we propose a two-step approach: first, learn IRs using all data, then fine-tune IRs with only labelled data to learn DRs, which can both utilize information from whole dataset and adapt the discriminative power to the given data. Extensive experiments and theoretical analysis demonstrate that SSL-C2ST outperforms traditional C2ST by effectively leveraging unlabeled data. We also offer a stronger empirically designed test achieving the SOTA performance in many two-sample testing datasets.

Nonparametric Feature Selection by Random Forests and Deep Neural Networks

Random forests are a widely used machine learning algorithm, but their computational efficiency is undermined when applied to large-scale datasets with numerous instances and useless features. Herein, we propose a nonparametric feature selection algorithm that incorporates random forests and deep neural networks, and its theoretical properties are also investigated under regularity conditions. Using different synthetic models and a real-world example, we demonstrate the advantage of the proposed algorithm over other alternatives in terms of identifying useful features, avoiding useless ones, and the computation efficiency. Although the algorithm is proposed using standard random forests, it can be widely adapted to other machine learning algorithms, as long as features can be sorted accordingly.