Optimal Multiscale Learning of Linear Operators
Jiaheng Chen, Daniel Sanz-Alonso
math.ST, math.NA, stat.ML
Submitted: 2026-06-15
Comments: 48 pages, 2 figures
License: http://creativecommons.org/licenses/by/4.0/
The gist: We study the statistical and computational limits of learning bounded linear operators between Sobolev spaces from noisy input-output data.
Terminology
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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