3/11/21 | 4:15pm | Online only

Anatoli Juditsky

Professor
University of Grenoble

Abstract: 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 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 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)

Bio: Anatoli Juditsky received the M.S.E.E. in 1985 from the Moscow Institute of Physics and Technology and the Ph.D. degree in Electrical Engineering in 1989 from the Institute of Control Sci., Moscow, USSR. Since 1999 he is Professor at the Department of Mathematics and Informatics (IM2AG) of the Université Grenoble Alpes France; he held a research position at INRIA-IRISA, Rennes, in 1991-1996, and then at INRIA, Grenoble, in 1996- 1999. His current research interests include stochastic and large-scale convex optimization, statistical theory and their applications.