Adaptive estimation from indirect observations

We discuss an approach to estimate aggregation and adaptive estimation based upon (nearly optimal) testing of convex hypotheses.
We show that in the situation where the observations stem from simple observation schemes (i.e., have Gaussian, discrete and Poisson distribution) and where the set of unknown signals is a finite union of convex and compact sets, the proposed approach leads to aggregation routines with nearly optimal performance. As an illustration, we consider application of the proposed estimates to the problem of recovery of unknown signal known to belong to a union of ellitopes in Gaussian observation scheme. The corresponding numerical routines can be implemented efficiently when the number of sets in the union is “not very large.” We illustrate the “practical performance” of the method in a numerical example of estimation in the single-index model.

Joint work with A. Nemirovski (Georgia Tech). https://arxiv.org/abs/2107.07836