Classification of sparse high-dimensional vectors

We consider a classication problem with high-dimensional vector samples. We observe $M$ samples drawn from $M$ populations and we want to classify a new vector $Z$. We suppose that the difference between the distributions of the populations is only in a shift that is a sparse vector. We obtain asymptotically (as the dimension $d$ tends to infinity) sharp classification boundary for the Gaussian noise and fixed sample size, and we propose classifiers that provide this boundary. [Joint work with Yuri Ingster]