# Covariance Selection Quality and Approximation Algorithms

Date: 2018-10-10 Add to Google CalendarTime: 1:30pm

Location: Holmes Hall 287

Speaker: Navid Tafaghodi Khajavi, EE PhD Candidate

Abstract:

The first problem we study in this work is the quality of statistical model selection. The statistical model selection often uses a distance measure such as the Kullback-Leibler (KL) divergence between the original distribution and the model distribution to quantify the quality of approximated model distribution. Although the KL divergence is minimized to obtain model approximation in many cases, there are other measures and divergences that can be used to do so. We extend the body of research by formulating the model approximation as a parametric detection problem between the original distribution and the model distribution. The proposed detection framework results in the computation of symmetric closeness measures such as receiver operating characteristic (ROC) and area under the curve (AUC). In the case of covariance selection, closeness measures such as KL divergence, reverse KL divergence, ROC, and AUC depend on the eigenvalues of the correlation approximation matrix (CAM). We find expressions for the KL divergence, the log-likelihood ratio, and the AUC as a function of the CAM. We present a simple integral to compute the AUC to assess the quality of an approximated model.

The second problem we target in this work is to formulate a general framework and approximation algorithms. We develop a multistage framework for graphical model approximation using a cascade of models such as trees. In particular, we look at the model approximation problem for Gaussian distributions as linear transformations of tree models. This is a new way to decompose the covariance matrix. Here, we propose an algorithm which incorporates the Cholesky factorization method to compute the decomposition matrix and thus can approximate a simple graphical model using a cascade of the Cholesky factorization of the tree approximation transformations. The overall graph is a cascade of factor graphs with each factor graph being a tree. Here, we present theoretical results that guarantee the convergence of the proposed model approximation using the cascade of tree decompositions.

<back>