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We derive the form of the variance-covariance matrix for any affine equivariant matrix-valued statistics when sampling from complex elliptical distributions. We then use this result to derive the variance-covariance matrix of the sample covariance matrix (SCM) as well as its theoretical mean squared error (MSE) when finite fourth-order moments exist. Finally, illustrative examples of the formulas are presented.

Publisher Copyright: Author 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.

In recent years, the use of Riemannian geometry has reportedly shown an increased performance for machine learning problems whose features lie in the symmetric positive definite (SPD) manifold. The present paper aims at reviewing several approaches based on this paradigm and provide a reproducible comparison of their output on a classic learning task of pedestrian detection. Notably, the robustness of these approaches to corrupted data will be assessed.

The theory of characteristic modes (TCMs) based on the surface integral equation (SIE) formalism is presented for the analysis of plasmonic nanostructures. With TCM, excitation-independent characteristic solutions of the underlying electromagnetic field problem are obtained. Both single and multiple particle nanostructures are analyzed using the conventional SIE approach and the TCM formalism. By comparing these results, the modes that are responsible of the observed scattering and absorption characteristics can be identified. These solutions can be further utilized in the design and tuning nanoantenna configurations for specific application purposes.

The paper considers the problem of estimating the covariance matrices of multiple classes in a low sample support condition, where the data dimensionality is comparable to, or larger than, the sample sizes of the available data sets. In such conditions' a common approach is to shrink the class sample covariance matrices (SCMs) towards the pooled SCM. The success of this approach hinges upon the ability to choose the optimal regularization parameter. Typically, a common regularization level is shared among the classes and determined via a procedure based on cross-validation. We use class-specific regularization levels since this enables the derivation of the optimal regularization parameter for each class in terms of the minimum mean squared error (MMSE). The optimal parameters depend on the true unknown class population covariances. Consistent estimators of the parameters can, however, be easily constructed under the assumption that the class populations follow (unspecified) elliptically symmetric distributions. We demonstrate the performance of the proposed method via a simulation study as well as via an application to discriminant analysis using both synthetic and real data sets.

The theory of characteristic modes (TCM) is applied to analyze optical properties of plasmonic nanoparticles. TCM gives excitation independent characteristic solutions of the electromagnetic problem formulated with surface integral equations and can provide additional information of the plasmonic resonances that can be difficult to obtain with conventional electromagnetic field analysis methods.

This paper studies how deforming the shape of a patch antenna affects its characteristic modes by applying the characteristic modes theory to analyze EM fields in antenna designs. The resonant frequencies of the lowest modes are in special focus in order to excite several modes in as small a space as possible.