Reconstruction of frequency-localized functions from pointwise samples via least squares and deep learning
This work addresses a fundamental signal processing problem for researchers and practitioners, but it is incremental as it builds on existing approximation-theoretic frameworks.
The paper tackles the problem of recovering frequency-localized functions from pointwise samples by analyzing least squares and deep learning methods, establishing recovery theorems with explicit bandwidth dependencies and providing numerical comparisons in one- and two-dimensional cases.
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.