Adaptive Estimation from Indirect Observations

We consider the problem of minimax and adaptive estimation of a signal assumed to belong to the union of convex sets. We show that the minimax risk in this problem (when measured in some norm or semi-norm) is similar (up to “logarithmic factors”) to the worst minimax risk of estimation over pairs of sets. Then, we examine the estimation algorithm which relies upon pairwise aggregation of estimators utilizing near-optimal testing of convex hypotheses. We show that when minimax estimators over “individual sets” are available, the proposed procedure yields an estimate with nearly minimax (up to “logarithmic factors”) performance. The construction of the corresponding aggregation procedures reduces to solving convex optimization problems and can be implemented efficiently. We also discuss a closely related question of the possibility of adaptive estimation over unions of convex sets. Finally, we consider signal estimation in Gaussian noise over unions of ellitopes (roughly, convex sets with “quadratic faces”) and show how minimax and adaptive estimates can be constructed in this problem. Joint work with Goldenshluger, A. (Haifa University) and A. Nemirovski (Georgia Tech)