  We develop a method for estimating well-conditioned and sparse covariance and
inverse covariance matrices from a sample of vectors drawn from a sub-gaussian
distribution in high dimensional setting. The proposed estimators are obtained
by minimizing the quadratic loss function and joint penalty of `1 norm and
variance of its eigenvalues. In contrast to some of the existing methods of
covariance and inverse covariance matrix estimation, where often the interest
is to estimate a sparse matrix, the proposed method is flexible in estimating
both a sparse and well-conditioned covariance matrix simultaneously. The
proposed estimators are optimal in the sense that they achieve the minimax rate
of estimation in operator norm for the underlying class of covariance and
inverse covariance matrices. We give a very fast algorithm for computation of
these covariance and inverse covariance matrices which is easily scalable to
large scale data analysis problems. The simulation study for varying sample
sizes and variables shows that the proposed estimators performs better than
several other estimators for various choices of structured covariance and
inverse covariance matrices. We also use our proposed estimator for tumor
tissues classification using gene expression data and compare its performance
with some other classification methods.
