The Cost of Discretization in Functional Linear Regression: Minimax Rates and Adaptation

What the math gives to ML

The paper gives a resolution-aware bias–variance principle for learning from noisy discretized functions: increasing the number of trajectories and increasing the number of samples per trajectory provide different kinds of information. Its transferable asset is the explicit separation between statistical error, amplified measurement noise, fixed-grid aliasing, and covariance-identification error, together with adaptive screening of spectral blocks. A practical neural-network adaptation is a Fourier or learned-spectral front end whose active mode budget is selected from dataset size, sampling resolution, and empirically estimated spectral decay rather than fixed in advance. This is most relevant to neural operators, sequence models for sampled signals, and models deployed across changing sensor resolutions.