提出无需限制权重的神经网络显著性检验新框架，更贴合实际深度学习。
Scale-Insensitive Neural Network Significance Tests
- 用Rademacher复杂度替代熵计算，摆脱权重有界假设。
- 仅需目标函数属于Sobolev空间，平滑性要求大幅降低。
- 基于矩约束构建筛空间，理论更契合现代深度学习实践。
本文提出一种尺度无关的神经网络显著性检验框架，通过三项关键创新实现对现有方法的重大推广。首先，以Rademacher复杂度界替代度量熵计算，使分析无需限定权重有界或特定架构。其次，将目标函数的正则性条件弱化为仅需属于Sobolev空间 $H^s([-1,1]^d)$ 且 $s > d/2$，在保持最优逼近率的同时显著放松光滑性假设。第三，引入基于矩约束而非权重约束的改进筛空间构造，为现代深度学习实践提供更自然的理论支撑。该方法在保持最优收敛速率的同时，建立了检验统计量的有效渐近分布。技术基础融合局部化理论、尖锐集中不等式及尺度无关的复杂度度量，可处理无界权重与一般Lipschitz激活函数。该框架在保持数学严谨性的同时，更贴近当代深度学习的实际应用。
原文摘要 · Abstract (English)
This paper develops a scale-insensitive framework for neural network significance testing, substantially generalizing existing approaches through three key innovations. First, we replace metric entropy calculations with Rademacher complexity bounds, enabling the analysis of neural networks without requiring bounded weights or specific architectural constraints. Second, we weaken the regularity conditions on the target function to require only Sobolev space membership $H^s([-1,1]^d)$ with $s > d/2$, significantly relaxing previous smoothness assumptions while maintaining optimal approximation rates. Third, we introduce a modified sieve space construction based on moment bounds rather than weight constraints, providing a more natural theoretical framework for modern deep learning practices. Our approach achieves these generalizations while preserving optimal convergence rates and establishing valid asymptotic distributions for test statistics. The technical foundation combines localization theory, sharp concentration inequalities, and scale-insensitive complexity measures to handle unbounded weights and general Lipschitz activation functions. This framework better aligns theoretical guarantees with contemporary deep learning practice while maintaining mathematical rigor.
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