改进带限函数置信区域构造，提升小样本稳定性与覆盖精度
Derandomizing Simultaneous Confidence Regions for Band-Limited Functions by Improved Norm Bounds and Majority-Voting Schemes
- 在Paley-Wiener再生核希尔伯特空间中，结合随机化霍夫丁与经验伯恩斯坦界
- 根据样本量和输入信息量自适应选择最优界，显著缩小置信区域
- 通过多数投票聚合子样本结果，保持联合覆盖率且提升稳定性
带限函数是系统理论与信号处理中的基础对象。本文针对从含噪输入输出测量中构建带限函数同时置信区域的非参数、非渐近方法进行改进，工作于Paley-Wiener再生核希尔伯特空间。通过小样本采用均匀随机化霍夫丁不等式，大样本使用经验伯恩斯坦界，收紧核范数上界。推导出一个基于样本量和输入信息量的近似阈值，决定采用哪种边界。最后应用多数投票策略对随机子样本的置信集进行聚合，提升了稳定性和置信区域大小。证明即使单个输入的聚合区间仍保持联合覆盖性。数值实验验证了这些改进的有效性。
原文摘要 · Abstract (English)
Band-limited functions are fundamental objects that are widely used in systems theory and signal processing. In this paper we refine a recent nonparametric, nonasymptotic method for constructing simultaneous confidence regions for band-limited functions from noisy input-output measurements, by working in a Paley-Wiener reproducing kernel Hilbert space. Kernel norm bounds are tightened using a uniformly-randomized Hoeffding's inequality for small samples and an empirical Bernstein bound for larger ones. We derive an approximate threshold, based on the sample size and how informative the inputs are, that governs which bound to deploy. Finally, we apply majority voting to aggregate confidence sets from random subsamples, boosting both stability and region size. We prove that even per-input aggregated intervals retain their simultaneous coverage guarantee. These refinements are also validated through numerical experiments.
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