Derandomizing Simultaneous Confidence Regions for Band-Limited Functions by Improved Norm Bounds and Majority-Voting Schemes

2025-06-21

AI Summary

This paper addresses nonparametric inference for bandlimited functions under noisy measurements, focusing on constructing simultaneous confidence regions. Building upon the Paley–Wiener reproducing kernel Hilbert space framework, we propose a non-asymptotic, data-adaptive method: first deriving a data-driven norm-bound threshold, then aggregating confidence sets from multiple random subsamples via majority voting. Theoretically, this ensemble strategy rigorously guarantees simultaneous coverage probability. Compared to existing approaches, it employs uniform random Hoeffding inequalities for small samples and empirical Bernstein bounds for large samples—yielding substantially tighter confidence regions while enhancing robustness against outliers and input sparsity. Numerical experiments demonstrate superior accuracy and stability across varying signal-to-noise ratios and sampling densities.

Technology Category

• Machine Learning: Kernel Methods
• Reasoning under Uncertainty: Stochastic Optimization
• Search and Optimization: Non-convex Optimization

Application Category

• Security and Privacy: Large-scale security measurements
• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods

Abstract

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.

Problem

• Constructing confidence regions for band-limited functions
• Improving norm bounds for small and large samples
• Boosting stability via majority-voting schemes

Innovation

• Tighter kernel norm bounds via randomized inequalities
• Empirical Bernstein bound for larger samples
• Majority voting for stable confidence aggregation