用最小二乘与深度学习从采样点重构频域局部函数，给出理论保证。
Reconstruction of frequency-localized functions from pointwise samples via least squares and deep learning
- 基于Slepian基的最小二乘法，在低维均匀采样下实现频带依赖的重建
- 证明了深度网络在特定架构与训练条件下可准确逼近带限函数
- 对比一维和二维函数重建，揭示理论与实际应用间的差距
从点采样中恢复频域局部函数是信号处理中的基本问题。本文从逼近论角度出发，研究最小二乘法与基于深度学习的方法。首先，针对低维均匀随机采样，建立基于Slepian基的最小二乘重构新定理，明确跟踪了带宽与采样复杂度之间的依赖关系。在此基础上，提出一个带限函数通过深度学习从点采样中重构的保证结果，该结果以实用存在性定理形式呈现，给出了网络结构、训练过程与数据获取的充分条件。为补充理论分析，我们对一维和二维函数的最小二乘与深度学习方法进行了数值比较。最后讨论了理论局限性及理论与实际实现间的实践差距。
原文摘要 · Abstract (English)
Recovering frequency-localized functions from pointwise data is a fundamental task in signal processing. We examine this problem from an approximation-theoretic perspective, focusing on least squares and deep learning-based methods. First, we establish a novel recovery theorem for least squares approximations using the Slepian basis from uniform random samples in low dimensions, explicitly tracking the dependence of the bandwidth on the sampling complexity. Building on these results, we then present a recovery guarantee for approximating bandlimited functions via deep learning from pointwise data. This result, framed as a practical existence theorem, provides conditions on the network architecture, training procedure, and data acquisition sufficient for accurate approximation. To complement our theoretical findings, we perform numerical comparisons between least squares and deep learning for approximating one- and two-dimensional functions. We conclude with a discussion of the theoretical limitations and the practical gaps between theory and implementation.
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