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We propose a novel application of the Simultaneous Orthogonal MatchingPursuit (S-OMP) procedure for sparsistant variable selection in ultra-highdimensional multi-task regression problems. Screening of variables, asintroduced in \cite{fan08sis}, is an efficient and highly scalable way toremove many irrelevant variables from the set of all variables, while retainingall the relevant variables. S-OMP can be applied to problems with hundreds ofthousands of variables and once the number of variables is reduced to amanageable size, a more computationally demanding procedure can be used toidentify the relevant variables for each of the regression outputs. To ourknowledge, this is the first attempt to utilize relatedness of multiple outputsto perform fast screening of relevant variables. As our main theoreticalcontribution, we prove that, asymptotically, S-OMP is guaranteed to reduce anultra-high number of variables to below the sample size without losing truerelevant variables. We also provide formal evidence that a modified Bayesianinformation criterion (BIC) can be used to efficiently determine the number ofiterations in S-OMP. We further provide empirical evidence on the benefit ofvariable selection using multiple regression outputs jointly, as opposed toperforming variable selection for each output separately. The finite sampleperformance of S-OMP is demonstrated on extensive simulation studies, and on agenetic association mapping problem. $Keywords$ Adaptive Lasso; Greedy forwardregression; Orthogonal matching pursuit; Multi-output regression; Multi-tasklearning; Simultaneous orthogonal matching pursuit; Sure screening; Variableselection

Ultra-high Dimensional Multiple Output Learning With Simultaneous Orthogonal Matching Pursuit: A Sure Screening Approach