This paper addresses the problem of inverse covariance (also known as precision matrix) estimation in high-dimensional settings. Specifically, we focus on two classes of estimators: linear shrinkage estimators with a target proportional to the identity matrix, and estimators derived from data augmentation (DA). Here, DA refers to the common practice of enriching a dataset with artificial samples—typically generated via a generative model or through random transformations of the original data—prior to model fitting. For both classes of estimators, we derive estimators and provide concentration bounds for their quadratic error. This allows for both method comparison and hyperparameter tuning, such as selecting the optimal proportion of artificial samples. On the technical side, our analysis relies on tools from random matrix theory. We introduce a novel deterministic equivalent for generalized resolvent matrices, accommodating dependent samples with specific structure. We support our theoretical results with numerical experiments.

Shannon's mutual information of a random multiple antenna and multipath time varying channel is studied in the general case where the process constructed from the channel coefficients is an ergodic and stationary process which is assumed to be available at the receiver. From this viewpoint, the channel can also be represented by an ergodic self-adjoint block-Jacobi operator, which is close in many aspects to a block version of a random Schrödinger operator. The mutual information is then related to the so-called density of states of this operator. In this paper, it is shown that under the weakest assumptions on the channel, the mutual information can be expressed in terms of a matrix-valued stochastic process coupled with the channel process. This allows numerical approximations of the mutual information in this general setting. Moreover, assuming further that the channel coefficient process is a Markov process, a representation for the mutual information offset in the large signal to noise ratio regime is obtained in terms of another related Markov process. This generalizes previous results from Levy et al.. It is also illustrated how the mutual information expressions that are closely related to those predicted by the random matrix theory can be recovered in the large dimensional regime.