Strong correlations between explanatory variables are problematic for high-dimensional regularized regression methods. Due to the violation of the Irrepresentable Condition, the popular LASSO method may suffer from false inclusions of inactive variables. In this paper, we propose pre-processing with orthogonal decompositions (PROD) for the explanatory variables in high-dimensional regressions. The PROD procedure is constructed based upon a generic orthogonal decomposition of the design matrix. We demonstrate by two concrete cases that the PROD approach can be effectively constructed for improving the performance of high-dimensional penalized regression. Our theoretical analysis reveals their properties and benefits for high-dimensional penalized linear regression with LASSO. Extensive numerical studies with simulations and data analysis show the promising performance of the PROD.
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