Optimal Multiscale Learning of Linear Operators

2026-06-15

AI Summary

This work addresses the statistical and computational challenges of learning bounded linear operators between Sobolev spaces from noisy data. The authors reformulate operator learning as an infinite-dimensional matrix regression in wavelet coordinates and propose a finite-resolution, block-wise least squares estimator that exploits multiscale structure. To account for the heterogeneous local estimation difficulty across scales, they introduce a scale-adaptive sampling strategy. Under the Sobolev operator norm loss, the proposed method achieves the minimax optimal convergence rate while attaining the best possible computational complexity among dense least squares–type algorithms.

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.

Problem

• linear operators
• Sobolev spaces
• multiscale learning
• minimax rates
• noisy data

Innovation

• multiscale learning
• linear operators
• wavelet coordinates
• minimax rates
• scale-adaptive estimation