Ultra-high Dimensional Multiple Output Learning With Simultaneous Orthogonal Matching Pursuit: Screening Approach
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We propose a novel application of the Simultaneous Orthogonal Matching Pursuit (SOMP) procedure to perform variable selection in ultra-high dimensional multiple output regression problems, which is the first attempt to utilize multiple outputs to perform fast removal of the irrelevant variables. As our main theoretical contribution, we show that the S-OMP can be used to reduce an ultra-high number of variables to below the sample size, without losing relevant variables. We also provide formal evidence that the modified Bayesian information criterion (BIC) can be used to efficiently select the number of iterations in the S-OMP. Once the number of variables has been reduced to a manageable size, we show that a more computationally demanding procedure can be used to identify the relevant variables for each of the regression outputs. We further provide evidence on the benefit of variable selection using the regression outputs jointly, as opposed to performing variable selection for each output separately. The finite sample performance of the S-OMP has been demonstrated on extensive simulation studies.