The paper compares the Lasso and Dantzig selector in high-dimensional regression under a sparsity assumption. Both methods are shown to exhibit similar behavior in terms of prediction risk and estimation loss. The Lasso estimator minimizes the sum of squared residuals with an $\ell_1$ penalty, while the Dantzig selector satisfies a constraint on the maximum $\ell_\infty$ norm of the residual vector. The authors derive oracle inequalities for both methods in nonparametric regression and linear models, showing that they achieve similar performance in terms of prediction loss and $\ell_p$ estimation loss for $1 \leq p \leq 2$. The results are non-asymptotic and hold under general assumptions on the design matrix. The paper also discusses the computational feasibility of the Dantzig selector, which reduces to a linear programming problem. The main findings include the approximate equivalence of the Lasso and Dantzig selectors in terms of prediction loss and the derivation of sparsity oracle inequalities for both methods. The results are presented in the context of both linear and nonparametric regression models.The paper compares the Lasso and Dantzig selector in high-dimensional regression under a sparsity assumption. Both methods are shown to exhibit similar behavior in terms of prediction risk and estimation loss. The Lasso estimator minimizes the sum of squared residuals with an $\ell_1$ penalty, while the Dantzig selector satisfies a constraint on the maximum $\ell_\infty$ norm of the residual vector. The authors derive oracle inequalities for both methods in nonparametric regression and linear models, showing that they achieve similar performance in terms of prediction loss and $\ell_p$ estimation loss for $1 \leq p \leq 2$. The results are non-asymptotic and hold under general assumptions on the design matrix. The paper also discusses the computational feasibility of the Dantzig selector, which reduces to a linear programming problem. The main findings include the approximate equivalence of the Lasso and Dantzig selectors in terms of prediction loss and the derivation of sparsity oracle inequalities for both methods. The results are presented in the context of both linear and nonparametric regression models.