Точная минимаксная и адаптивная селекция в моделях с разреженными переменными

In this talk, we discuss the problem of variable selection in the Gaussian sequence model in $\mathbb{R}^d$ for classes of $s$-sparse vectors separated from zero by a positive constant $a$. In some cases, using expected Hamming loss, we find explicitly the minimax selectors and obtain exact expressions for the non-asymptotic minimax risk as a function of $d,s$, and $a$. The obtained results are extended to dependent or non-Gaussian observations. Similar conclusions are derived for the probability of wrong recovery of a sparsity pattern. We also establish necessary and sufficient conditions for the possibility of almost full and exact variable selection (asymptotically). Moreover, we propose data-driven selectors that provide almost full and exact variable selection adaptively in the parameters of the classes. This is joint work with Cristina Butucea (France) and Alexandre Tsybakov (France).