In this work, we propose a new regularization approach for linear least-squares problems with random matrices. In
the proposed constrained perturbation regularization approach, an artificial perturbation matrix with a bounded norm is forced
into the system model matrix. This perturbation is introduced to improve the singular-value structure of the model matrix and,
hence, the solution of the estimation problem. Relying on the randomness of the model matrix, a number of deterministic equivalents from random matrix theory are applied to derive the near-optimum regularizer that minimizes the mean-squared error of the estimator. Simulation results demonstrate that the proposed approach outperforms a set of benchmark regularization methods for various estimated signal characteristics. In addition, simulations show that our approach is robust in the presence of model uncertainty.