Regression graphs and sparsity-inducing reparametrizations

Motivated by the important statistical role of sparsity, the paper uncovers four reparametrizations for covariance matrices in which sparsity is associated with conditional independence graphs in a notional Gaussian model. The intimate relationship between the Iwasawa decomposition of the general linear group and the open cone of positive definite matrices allows a unifying perspective. Specifically, the positive definite cone can be reconstructed without loss or redundancy from the exponential map applied to four Lie subalgebras determined by the Iwasawa decomposition of the general linear group. This accords geometric interpretations to the reparametrizations and the corresponding notion of sparsity. Conditions that ensure legitimacy of the reparametrizations for statistical models are identified. While the focus of this work is on understanding population-level structure, there are strong methodological implications. In particular, since the population-level sparsity manifests in a vector space, imposition of sparsity on relevant sample quantities produces a covariance estimate that respects the positive definite cone constraint.