Abstract
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.
Highlights
A N increasingly common scenario in modern supervised learning problems is that the dimension p of the data is large compared to the number of available training samples n or exceed it multifold (p n)
We address the problem of high-dimensional covariance matrix estimation in a multiclass setup, where there are K different classes or populations, each comprising nk, k = 1, . . . , K, independent and identically distributed (i.i.d.) p-dimensional samples
Two regularized sample covariance matrix estimators specified in (4) and in (9) for multiclass problems were considered in this work and their theoretically optimal class-specific tuning parameters were derived in Theorem 1 and Theorem 3, respectively
Summary
A N increasingly common scenario in modern supervised learning problems is that the dimension p of the data is large compared to the number of available training samples n or exceed it multifold (p n). The conventional estimate for the covariance matrix is the unbiased sample covariance matrix (SCM) defined for class k by nk. Better estimators can be developed by using regularization, where the key idea is to shift or shrink the estimator toward a predetermined target or model This can significantly decrease the variance of the estimator and improve the overall performance by reducing its mean squared error (MSE). In [20], a linear multi-target shrinkage covariance matrix estimator was proposed, which optimizes the tuning parameters using low-complexity leaveone-out cross-validation. Unif(a, b) denote the discrete and continuous univariate uniform distributions on set {a, a + 1, . . . , b − 1, b} and on the open interval (a, b), respectively
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