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Least squares fitting is in general not useful for high-dimensional linearmodels, in which the number of predictors is of the same or even larger orderof magnitude than the number of samples. Theory developed in recent years hascoined a paradigm according to which sparsity-promoting regularization isregarded as a necessity in such setting. Deviating from this paradigm, we showthat non-negativity constraints on the regression coefficients may be similarlyeffective as explicit regularization if the design matrix has additionalproperties, which are met in several applications of non-negative least squares(NNLS). We show that for these designs, the performance of NNLS with regard toprediction and estimation is comparable to that of the lasso. We argue furtherthat in specific cases, NNLS may have a better $\ell_{\infty}$-rate inestimation and hence also advantages with respect to support recovery whencombined with thresholding. From a practical point of view, NNLS does notdepend on a regularization parameter and is hence easier to use.

Non-negative least squares for high-dimensional linear models: consistency and sparse recovery without regularization