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.

翻译：我们研究了从带噪输入-输出数据中学习Sobolev空间之间有界线性算子的统计与计算极限。在小波坐标系下，该问题被重新表述为一个具有异质双边多尺度结构的无限维矩阵回归问题。我们在Sobolev算子范数损失下建立了极小极大速率，并构造了一个达到该速率的有限分辨率分块最小二乘估计量。分析揭示了不同尺度间非均匀的局部估计难度，这一性质可在算法层面加以利用：通过分配尺度自适应样本量，该估计量在所有稠密最小二乘实现中达到了最优计算代价。