The estimation of covariance matrices of multiple classes with limited training data is a difficult problem. The sample covariance matrix (SCM) is known to perform poorly when the number of variables is large compared to the available number of samples. In order to reduce the mean squared error (MSE) of the SCM, regularized (shrinkage) SCM estimators are often used. In this work, we consider regularized SCM (RSCM) estimators for multiclass problems that couple together two different target matrices for regularization: the pooled (average) SCM of the classes and the scaled identity matrix. Regularization toward the pooled SCM is beneficial when the population covariances are similar, whereas regularization toward the identity matrix guarantees that the estimators are positive definite. We derive the MSE optimal tuning parameters for the estimators as well as propose a method for their estimation under the assumption that the class populations follow (unspecified) elliptical distributions with finite fourth-order moments. The MSE performance of the proposed coupled RSCMs are evaluated with simulations and in a regularized discriminant analysis (RDA) classification set-up on real data. The results based on three different real data sets indicate comparable performance to cross-validation but with a significant speed-up in computation time.

中文翻译：

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多类耦合正则化样本协方差矩阵估计器


在有限的训练数据下估计多类协方差矩阵是一个难题。众所周知，当变量数量与可用样本数量相比较大时，样本协方差矩阵 (SCM) 的性能会很差。为了减少 SCM 的均方误差 (MSE)，经常使用正则化（收缩）SCM 估计器。在这项工作中，我们考虑用于多类问题的正则化 SCM (RSCM) 估计器，将两个不同的目标矩阵耦合在一起以进行正则化：类的汇集（平均）SCM 和缩放的单位矩阵。当总体协方差相似时，针对合并 SCM 的正则化是有益的，而针对单位矩阵的正则化可保证估计量是正定的。我们推导了估计器的 MSE 最优调整参数，并提出了一种估计方法，假设类总体遵循（未指定的）具有有限四阶矩的椭圆分布。所提出的耦合 RSCM 的 MSE 性能通过模拟和真实数据的正则判别分析 (RDA) 分类设置进行评估。基于三个不同真实数据集的结果表明，其性能与交叉验证相当，但计算时间显着加快。