טכניון מכון טכנולוגי לישראל
הטכניון מכון טכנולוגי לישראל - בית הספר ללימודי מוסמכים  
M.Sc Thesis
M.Sc StudentBloom Noam
SubjectCovariance Matrix Estimation Under a Combined Low-Rank
and graphical Model Structure
DepartmentDepartment of Electrical Engineering
Supervisor Professor Talmon Ronen
Full Thesis textFull thesis text - English Version


Abstract

The problem of covariance matrix estimation involves estimating it given a sample of data points. It is of paramount importance in many fields, owing to the fundamental nature of second moments in probability and statistics. Although it has its roots early in the last century, in recent years it has attracted considerable renewed interest from both theoretical and practical standpoints. Two prevalent types of data models are inherently related to the covariance matrix and are therefore of special interest in estimation problems. The first of these is a latent linear factor model, also strongly related to PCA, which is manifested as a low rank component in the covariance matrix. The second is undirected graphical model structure (Markov random field), which is related to presence of zeros in the inverse covariance (precision) matrix. In this work we consider the problem of estimating a covariance matrix subject to both types of models simultaneously. We demonstrate the attractiveness of the combined model and propose a novel estimator to address it. We show that our estimator outperforms other alternatives on both synthetic and real-world data, explore a few of its theoretical properties such as consistency, and develop a fast algorithm for solving the associated optimization problem.