Semi-Supervised U-statistics

• URL: http://arxiv.org/abs/2402.18921v2
• Date: Sat, 9 Mar 2024 07:16:46 GMT
• Title: Semi-Supervised U-statistics
• Authors: Ilmun Kim, Larry Wasserman, Sivaraman Balakrishnan, Matey Neykov
• Abstract summary: We introduce semi-supervised U-statistics enhanced by the abundance of unlabeled data. We show that the proposed approach exhibits notable efficiency gains over classical U-statistics. We propose a refined approach that outperforms the classical U-statistic across all degeneracy regimes.
• Score: 22.696630428733204
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: Semi-supervised datasets are ubiquitous across diverse domains where obtaining fully labeled data is costly or time-consuming. The prevalence of such datasets has consistently driven the demand for new tools and methods that exploit the potential of unlabeled data. Responding to this demand, we introduce semi-supervised U-statistics enhanced by the abundance of unlabeled data, and investigate their statistical properties. We show that the proposed approach is asymptotically Normal and exhibits notable efficiency gains over classical U-statistics by effectively integrating various powerful prediction tools into the framework. To understand the fundamental difficulty of the problem, we derive minimax lower bounds in semi-supervised settings and showcase that our procedure is semi-parametrically efficient under regularity conditions. Moreover, tailored to bivariate kernels, we propose a refined approach that outperforms the classical U-statistic across all degeneracy regimes, and demonstrate its optimality properties. Simulation studies are conducted to corroborate our findings and to further demonstrate our framework.

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