Convergence Rates for Learning Linear Operators from Noisy Data

This paper studies the learning of linear operators between infinite-dimensional Hilbert spaces. The training data comprises pairs random input vectors in a space and their noisy images under an unknown self-adjoint operator. Assuming that operator is diagonalizable known basis, this work solves equivalent inverse problem estimating operator’s eigenvalues given data. Adopting Bayesian approach, theoretical analysis establishes posterior contraction rates infinite limit with Gaussian priors are not directly linked to forward map problem. main results also include learning-theoretic generalization error guarantees for wide range distribution shifts. These convergence quantify effects smoothness true eigenvalue decay or growth, compact unbounded operators, respectively, on sample complexity. Numerical evidence supports theory diagonal nondiagonal settings.