A Well-Conditioned and Sparse Estimation of Covariance and Inverse Covariance Matrices Using a Joint Penalty

摘要

We develop a method for estimating well-conditioned and sparsecovariance and inverse covariance matrices from a sample ofvectors drawn from a sub-Gaussian distribution in highdimensional setting. The proposed estimators are obtained byminimizing the quadratic loss function and joint penalty of$ell_1$ norm and variance of its eigenvalues. In contrast tosome of the existing methods of covariance and inversecovariance matrix estimation, where often the interest is toestimate a sparse matrix, the proposed method is flexible inestimating both a sparse and well-conditioned covariance matrixsimultaneously. The proposed estimators are optimal in the sensethat they achieve the mini-max rate of estimation in operatornorm for the underlying class of covariance and inversecovariance matrices. We give a very fast algorithm forcomputation of these covariance and inverse covariance matriceswhich is easily scalable to large scale data analysis problems.The simulation study for varying sample sizes and variablesshows that the proposed estimators performs better than severalother estimators for various choices of structured covarianceand inverse covariance matrices. We also use our proposedestimator for tumor tissues classification using gene expressiondata and compare its performance with some other classificationmethods. color="gray">