This paper introduces and analyzes a method for regularizing covariance matrices using hard thresholding. The authors show that the thresholded estimate is consistent in the operator norm as long as the true covariance matrix is sparse, the variables are Gaussian or sub-Gaussian, and $(\log p)/n \to 0$, providing explicit rates of convergence. The results are uniform over families of covariance matrices that satisfy a natural notion of sparsity. The paper discusses a resampling scheme for threshold selection and proves a general cross-validation result. It also compares thresholding to other covariance estimators in simulations and an example from climate data. The main contributions include: 1. **Consistency of Thresholding**: The thresholded estimate is consistent in the operator norm for a wide class of sparse covariance matrices, provided $(\log p)/n \to 0$. 2. **Cross-Validation for Threshold Selection**: A novel cross-validation method is proposed to select the threshold, which is theoretically justified. 3. **Comparison with Other Methods**: The performance of thresholding is compared with banding and other permutation-invariant estimators, showing its advantages in certain scenarios. 4. **Application to Climate Data**: The method is applied to climate data, demonstrating its effectiveness in separating spatial patterns. The paper provides a comprehensive theoretical and practical framework for using thresholding as a regularization technique for covariance matrices, making it a valuable contribution to the field of high-dimensional statistics.This paper introduces and analyzes a method for regularizing covariance matrices using hard thresholding. The authors show that the thresholded estimate is consistent in the operator norm as long as the true covariance matrix is sparse, the variables are Gaussian or sub-Gaussian, and $(\log p)/n \to 0$, providing explicit rates of convergence. The results are uniform over families of covariance matrices that satisfy a natural notion of sparsity. The paper discusses a resampling scheme for threshold selection and proves a general cross-validation result. It also compares thresholding to other covariance estimators in simulations and an example from climate data. The main contributions include: 1. **Consistency of Thresholding**: The thresholded estimate is consistent in the operator norm for a wide class of sparse covariance matrices, provided $(\log p)/n \to 0$. 2. **Cross-Validation for Threshold Selection**: A novel cross-validation method is proposed to select the threshold, which is theoretically justified. 3. **Comparison with Other Methods**: The performance of thresholding is compared with banding and other permutation-invariant estimators, showing its advantages in certain scenarios. 4. **Application to Climate Data**: The method is applied to climate data, demonstrating its effectiveness in separating spatial patterns. The paper provides a comprehensive theoretical and practical framework for using thresholding as a regularization technique for covariance matrices, making it a valuable contribution to the field of high-dimensional statistics.