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We observe a $N\times M$ matrix of independent, identically distributedGaussian random variables which are centered except for elements of somesubmatrix of size $n\times m$ where the mean is larger than some $a>0$. Thesubmatrix 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 significantlylarge 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 dosharp variable selection. We also prove the minimax lower bounds undernecessary conditions which are complementary to the previous conditions. Thecritical values $a^*$ separating the necessary and sufficient conditions aresharp (we show exact constants).  We note a gap between the critical values $a^*$ for selection of variablesand 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 noconsistent selector of the corresponding variables can be found.

Sharp Variable Selection of a Sparse Submatrix in a High-Dimensional Noisy Matrix