Conditional Minimum Volume Ellipsoid with Applications to Subset Selection for MVE Estimator and Multiclass Discrimination

In this paper, we present a new formulation for constructing an ellipsoid which generalizes the computation of the minimum volume covering ellipsoid, based on the CVaR minimization technique proposed by Rockafellar and Uryasev (2002). The proposed ellipsoid construction is formulated as a convex optimization and an interior point algorithm for the solution can be developed. In addition, the optimization gives an upper bound of the volume of the ellipsoid associated with the MVE robust estimator, which fact can be exploited for approximate computations of the estimator. Also, potential applicability of the new ellipsoid construction is discussed through two statistical problems: 1) robust statistics computations including outlier detection and the computation of the MVE estimator; 2) a multiclass discrimination problem, where the maximization of the normal likelihood function is characterized in the context of the ellipsoid construction. Numerical results are given, showing the nice computational efficiency of the proposed interior point algorithm and the capability of the proposed generalization.