Convergence Rates for Learning Linear Operators from Noisy Data
This work addresses the challenge of operator learning in functional analysis, providing theoretical foundations for applications in fields like PDEs and signal processing, though it is incremental in extending Bayesian methods to this context.
This paper tackles the problem of learning linear operators between infinite-dimensional Hilbert spaces from noisy data, establishing posterior contraction rates and generalization error guarantees that quantify the effects of data smoothness and eigenvalue decay or growth on sample complexity.
This paper studies the learning of linear operators between infinite-dimensional Hilbert spaces. The training data comprises pairs of random input vectors in a Hilbert space and their noisy images under an unknown self-adjoint linear operator. Assuming that the operator is diagonalizable in a known basis, this work solves the equivalent inverse problem of estimating the operator's eigenvalues given the data. Adopting a Bayesian approach, the theoretical analysis establishes posterior contraction rates in the infinite data limit with Gaussian priors that are not directly linked to the forward map of the inverse problem. The main results also include learning-theoretic generalization error guarantees for a wide range of distribution shifts. These convergence rates quantify the effects of data smoothness and true eigenvalue decay or growth, for compact or unbounded operators, respectively, on sample complexity. Numerical evidence supports the theory in diagonal and non-diagonal settings.