Population-level sparsity induced by reparametrisation

Oregon Convention Center 

That parametrisation and population-level sparsity are intrinsically linked is a fundamental point that has not been emphasised. In the particular context of covariance matrices, we address the following question: given a statistical problem, not obviously sparse in its natural formulation, can a sparsity-inducing reparametrisation be deduced? Four types of reparametrisation are initially considered, two old (Battey, 2017; Rybak and Battey, 2021) and two new, in which sparsity manifests in different vector spaces. We establish a result of sufficient generality to apply in the four cases, recovering known results and generating new ones. In particular, for the new parametrisations we uncover the structure induced on physically natural scales through sparsity on the transformed scale, and the converse result of interest: that matrices encoding such structure are sparse after reparametrisation. The richest of the four structures uncovered turns out to be that of the joint-response graphs studied by Wermuth and Cox (2004), and for these we provide an interpretation of the parameters in the new parametrisation. Unification of old and new parametrisations is provided through the so-called Iwasawa decomposition of the general linear group of $p$-dimensional invertible matrices. Since the structures emerge either from an $r+s=p$ partitioning of $p$, or from a $1+1+\cdots$ recursive partitioning of $p$, our analysis points to a class of further structures, between the two extremes, which are sparse after reparametrisation. These correspond to a general $r_1+\cdots+r_k=p$ recursive partitioning with $k\leq p$, whose interpretation is in terms of $k$ multivariate regression models. The granularity of the parametrisation thus determines the `locality' of the graphical models interpretation. There are direct methodological implications of the work owing to the manifestation of sparsity in a vector space, which evades awkward constraints on the parameter space from the positive definiteness requirement.
The talk is based on joint work with Karthik Bharath (University of Nottingham) and Jakub Rybak (Imperial College London).