  We observe a $N\times M$ matrix of independent, identically distributed
Gaussian random variables which are centered except for elements of some
submatrix of size $n\times m$ where the mean is larger than some $a>0$. The
submatrix is sparse in the sense that $n/N$ and $m/M$ tend to 0, whereas $n,\,
m, \, N$ and $M$ tend to infinity.
  We consider the problem of selecting the random variables with significantly
large mean values. We give sufficient conditions on $a$ as a function of $n,\,
m,\,N$ and $M$ and construct a uniformly consistent procedure in order to do
sharp variable selection. We also prove the minimax lower bounds under
necessary conditions which are complementary to the previous conditions. The
critical values $a^*$ separating the necessary and sufficient conditions are
sharp (we show exact constants).
  We note a gap between the critical values $a^*$ for selection of variables
and that of detecting that such a submatrix exists given by Butucea and Ingster
(2012). When $a^*$ is in this gap, consistent detection is possible but no
consistent selector of the corresponding variables can be found.
