Аннотация:
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).
|
