Cardinality-constrained feature selection for classification

We are concerned with the selection of a small subset of characteristics for which the classification of a system according to one of the states in its state set is optimal according to the Rayleigh quotient criterion. This problem is relevant in various scenarios where a few explanatory variables have to be selected from a large set, including sensor selection in sensor networks, classification in image processing, and feature selection in data mining for bioinformatics applications. We show that the optimization is equivalent to finding the submatrix of the features covariance matrix for which the sum of elements of its inverse is maximized, and we present bounds related to a similar metric based on elements of the original covariance matrix.