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Basis Pursuit (BP), Basis Pursuit DeNoising (BPDN), and LASSO are popularmethods for identifying important predictors in the high-dimensional linearregression model, i.e. when the number of rows of the design matrix X issmaller than the number of columns. By definition, BP uniquely recovers thevector of regression coefficients b if there is no noise and the vector b hasthe smallest L1 norm among all vectors s such that Xb=Xs (identifiabilitycondition). Furthermore, LASSO can recover the sign of b only under a muchstronger irrepresentability condition. Meanwhile, it is known that the modelselection properties of LASSO can be improved by hard-thresholding itsestimates. This article supports these findings by proving that thresholdedLASSO, thresholded BPDN and thresholded BP recover the sign of b in both thenoisy and noiseless cases if and only if b is identifiable and large enough. Inparticular, if X has iid Gaussian entries and the number of predictors growslinearly with the sample size, then these thresholded estimators can recoverthe sign of b when the signal sparsity is asymptotically below theDonoho-Tanner transition curve. This is in contrast to the regular LASSO, whichasymptotically recovers the sign of b only when the signal sparsity tends to 0.Numerical experiments show that the identifiability condition, unlike theirrepresentability condition, does not seem to be affected by the structure ofthe correlations in the $X$ matrix.

On the sign recovery by LASSO, thresholded LASSO and thresholded Basis Pursuit Denoising