Inverse learning in Hilbert scales
This provides theoretical convergence guarantees for inverse problems in statistical learning, but appears incremental as it extends existing regularization analysis to Hilbert scales.
The paper tackles the linear ill-posed inverse problem with noisy data in statistical learning, using general regularization schemes in Hilbert scales to derive convergence rates for approximate reconstructions. It establishes explicit error bounds for regression functions with smoothness given by source conditions.
We study the linear ill-posed inverse problem with noisy data in the statistical learning setting. Approximate reconstructions from random noisy data are sought with general regularization schemes in Hilbert scale. We discuss the rates of convergence for the regularized solution under the prior assumptions and a certain link condition. We express the error in terms of certain distance functions. For regression functions with smoothness given in terms of source conditions the error bound can then be explicitly established.