Regularity conditions play a pivotal role for sparse recovery in high-dimensional regression. In this paper, we present a weaker regularity condition and further discuss the relationships with other regularity conditions, such as restricted eigenvalue condition. We study the behavior of our new condition for design matrices with independent random columns uniformly drawn on the unit sphere. Moreover, the present paper shows that, under a sparsity scenario, the Lasso estimator and Dantzig selector exhibit similar behavior. Based on both methods, we derive, in parallel, more precise bounds for the estimation loss and the prediction risk in the linear regression model when the number of variables can be much larger than the sample size.