  We propose a novel application of the Simultaneous Orthogonal Matching
Pursuit (S-OMP) procedure for sparsistant variable selection in ultra-high
dimensional multi-task regression problems. Screening of variables, as
introduced in \cite{fan08sis}, is an efficient and highly scalable way to
remove many irrelevant variables from the set of all variables, while retaining
all the relevant variables. S-OMP can be applied to problems with hundreds of
thousands of variables and once the number of variables is reduced to a
manageable size, a more computationally demanding procedure can be used to
identify the relevant variables for each of the regression outputs. To our
knowledge, this is the first attempt to utilize relatedness of multiple outputs
to perform fast screening of relevant variables. As our main theoretical
contribution, we prove that, asymptotically, S-OMP is guaranteed to reduce an
ultra-high number of variables to below the sample size without losing true
relevant variables. We also provide formal evidence that a modified Bayesian
information criterion (BIC) can be used to efficiently determine the number of
iterations in S-OMP. We further provide empirical evidence on the benefit of
variable selection using multiple regression outputs jointly, as opposed to
performing variable selection for each output separately. The finite sample
performance of S-OMP is demonstrated on extensive simulation studies, and on a
genetic association mapping problem. $Keywords$ Adaptive Lasso; Greedy forward
regression; Orthogonal matching pursuit; Multi-output regression; Multi-task
learning; Simultaneous orthogonal matching pursuit; Sure screening; Variable
selection
