Optimal Multiscale Learning of Linear Operators

Jiaheng Chen; Daniel Sanz-Alonso

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arXiv · 2606.16913

Optimal Multiscale Learning of Linear Operators

Jiaheng Chen·Daniel Sanz-Alonso

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Abstract

We study the statistical and computational limits of learning bounded linear operators between Sobolev spaces from noisy input-output data. In wavelet coordinates, the problem is recast as an infinite-dimensional matrix regression problem with a heterogeneous two-sided multiscale structure. We establish minimax rates under Sobolev operator-norm loss and construct a finite-resolution blockwise least-squares estimator attaining these rates. The analysis reveals a nonuniform local estimation difficulty across scales, which can be exploited algorithmically: by assigning scale-adaptive sample sizes, the estimator achieves the optimal computational cost among dense least-squares implementations.

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Jiaheng Chen, Daniel Sanz-Alonso. 2026-06-15. Optimal Multiscale Learning of Linear Operators. https://arxiv.org/abs/2606.16913

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