Conditional Minimum Volume Ellipsoid with Applications to Subset Selection for MVE Estimator and Multiclass Discrimination
Conditional Minimum Volume Ellipsoid with Applications to Subset Selection for MVE Estimator and Multiclass Discrimination
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发表时间：
2006
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通讯作者：
Jun-ya Gotoh;A. Takeda
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作者：
Jun-ya Gotoh;A. Takeda
本文在Rockafellar和Uryasev（2002）提出的CVaR最小化技术的基础上，提出了一个构造椭球体的新公式，推广了椭球体最小体积覆盖的计算。所提出的椭球结构可表述为一个凸优化，并可开发求解的内点算法。此外，优化给出了与MVE鲁棒估计器相关的椭球体积的上界，这一事实可用于估计器的近似计算。并通过两个统计问题讨论了新椭球结构的潜在适用性：1)鲁棒统计计算，包括离群值检测和MVE估计量的计算；2)一个多类判别问题，其中在椭球构造的背景下描述了正态似然函数的最大值。数值结果表明，所提出的内点算法具有良好的计算效率和泛化能力。
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.