Multivariate Generalized Gaussian Distribution: Convexity and Graphical Models

TL;DR

This paper analyzes covariance estimation in multivariate generalized Gaussian distributions, demonstrating convexity properties and proposing a framework for structured estimation under sparsity constraints, applicable to various time series models.

Contribution

It introduces a geodesic convexity analysis for MGGD likelihood and a convex framework for structured covariance estimation with sparsity, broadening applicability.

Findings

Likelihood analysis based on geodesic convexity with weaker assumptions

Convex formulation for structured covariance estimation with sparsity constraints

Applicable to time-varying autoregressive processes and multivariate Laplace distributions

Abstract

We consider covariance estimation in the multivariate generalized Gaussian distribution (MGGD) and elliptically symmetric (ES) distribution. The maximum likelihood optimization associated with this problem is non-convex, yet it has been proved that its global solution can be often computed via simple fixed point iterations. Our first contribution is a new analysis of this likelihood based on geodesic convexity that requires weaker assumptions. Our second contribution is a generalized framework for structured covariance estimation under sparsity constraints. We show that the optimizations can be formulated as convex minimization as long the MGGD shape parameter is larger than half and the sparsity pattern is chordal. These include, for example, maximum likelihood estimation of banded inverse covariances in multivariate Laplace distributions, which are associated with time varying autoregressive processes.