On a dense minimizer of empirical risk in inverse problems

Properties of estimators of a functional parameter in an inverse problem setup are studied. We focus on estimators obtained through dense minimization (as opposed to minimization over δ-nets) of suitably defined empirical risk. At the cost of imposition of a sort of local finite-dimensionality assumption, we fill some gaps in the proofs of results published by Klemela and Mammen [Ann. Statist. 38 (2010), 482-511]. We also give examples of functional classes that satisfy the modified assumptions.