Multivariate Regression on High-Dimensional Networks

Abstract

In the modern world, many forms of statistical data models are high-dimensional in nature and can often be represented through a Markovian conditional dependence graph upon a sparse adjacency matrix, where sparsity manifests itself through a high proportion of zero-valued off-diagonal elements. Classical likelihood-based approaches do not account for sparsity well, thus requiring some form of heuristic or penalty. A multivariate Gaussian graphical model is a standard choice for continuous data, where a LASSO one-norm penalty can be employed to shrink the off-diagonal precision matrix elements. However, a complication arises when the distribution is derived from potentially sparse covariates, on an observation level or on a node level, and further, there may be multiple instances of the model, as adjacent samples, each having their own parameter set. As the main proposal, a Fused-LASSO-based estimator is presented to additionally encourage parameter similarity across such adjacent samples, utilising the conditional observation-level covariate dependence structure as a regression. Fortunately, through a novel block-matrix formulation, it is shown that the conditional model can be transformed and estimated via the Generalised LASSO. Both theoretical and computational results show that the prescribed optimisation offers accurate shrinkage for different dimensionalities and sample sizes, where increasing the latter also demonstrates asymptotic selection consistency when adapting the penalty weights. As a second recommendation, a novel dimensionality-reduction constraint is imposed on a multivariate Gaussian likelihood in order to take into account covariate influences when they are on a node level. This document aims to outline theoretical underpinnings including asymptotic results, algorithm recommendations for estimation, outputs from a simulation study, and an application example.

Date of Award	25 Jun 2025
Original language	English
Awarding Institution	University of Bath
Supervisor	Sandipan Roy (Supervisor) & Vangelis Evangelou (Supervisor)