The paper "The Sparsity and Bias of the LASSO Selection in High-Dimensional Linear Regression" by Cun-Hui Zhang and Jian Huang explores the properties of the LASSO (Least Absolute Shrinkage and Selection Operator) in high-dimensional linear regression models. The authors build upon previous work by Meinshausen and Buhlmann, Zhao and Yu, who established the consistency of the LASSO under certain conditions, such as the neighborhood stability condition and the strong irrepresentable condition. The main contributions of this paper are: 1. **Rate Consistency**: The authors provide conditions under which the LASSO is rate-consistent in terms of sparsity and bias. Specifically, they prove that the LASSO selects a model with the correct order of dimensionality and controls the bias at a level determined by the contributions of small regression coefficients and threshold bias. 2. **Sparsity Condition**: They introduce a more general concept of sparsity, where a model is sparse if most coefficients are small, rather than requiring all nonzero coefficients to be uniformly bounded away from zero. 3. **Sparse Riesz Condition**: Under a sparse Riesz condition on the correlation of design variables, the authors prove that the LASSO selects all variables with coefficients above a threshold determined by the controlled bias of the selected model. 4. **Convergence Rates**: The rate consistency of the LASSO selection implies that the sum of error squares for the mean response and the $\ell_p$-loss for the regression coefficients converge at optimal rates under certain conditions. The paper also discusses the implications of these results for the convergence rate of the LASSO estimator and provides sufficient conditions for the sparse Riesz condition for both deterministic and random design matrices. The authors conclude with a discussion of related work and further remarks on the practical implications of their findings.The paper "The Sparsity and Bias of the LASSO Selection in High-Dimensional Linear Regression" by Cun-Hui Zhang and Jian Huang explores the properties of the LASSO (Least Absolute Shrinkage and Selection Operator) in high-dimensional linear regression models. The authors build upon previous work by Meinshausen and Buhlmann, Zhao and Yu, who established the consistency of the LASSO under certain conditions, such as the neighborhood stability condition and the strong irrepresentable condition. The main contributions of this paper are: 1. **Rate Consistency**: The authors provide conditions under which the LASSO is rate-consistent in terms of sparsity and bias. Specifically, they prove that the LASSO selects a model with the correct order of dimensionality and controls the bias at a level determined by the contributions of small regression coefficients and threshold bias. 2. **Sparsity Condition**: They introduce a more general concept of sparsity, where a model is sparse if most coefficients are small, rather than requiring all nonzero coefficients to be uniformly bounded away from zero. 3. **Sparse Riesz Condition**: Under a sparse Riesz condition on the correlation of design variables, the authors prove that the LASSO selects all variables with coefficients above a threshold determined by the controlled bias of the selected model. 4. **Convergence Rates**: The rate consistency of the LASSO selection implies that the sum of error squares for the mean response and the $\ell_p$-loss for the regression coefficients converge at optimal rates under certain conditions. The paper also discusses the implications of these results for the convergence rate of the LASSO estimator and provides sufficient conditions for the sparse Riesz condition for both deterministic and random design matrices. The authors conclude with a discussion of related work and further remarks on the practical implications of their findings.