Sparsity induced by covariance transformation: some deterministic and probabilistic results

Abstract

Motivated by statistical challenges arising in modern scientific fields, notably genomics, this paper seeks embeddings in which relevant covariance models are sparse. The work exploits a bijective mapping between a strictly positive definite matrix and its orthonormal eigen-decomposition, and between an orthonormal eigenvector matrix and its principle matrix logarithm. This leads to a representation of covariance matrices in terms of skew-symmetric matrices, for which there is a natural basis representation, and through which sparsity is conveniently explored. This theoretical work establishes the possibility of exploiting sparsity in the new parametrization and converting the conclusion back to the one of interest, a prospect of high relevance in statistics. The statistical aspects associated with this operation, while not a focus of the present work, are briefly discussed.