Contributions to High-dimensional Data Analysis using Factor Models and Low Rank Approximations.

Abstract

Modern information technology has enabled collecting data of unprecedented size and complexity, but it also presents significant challenges to learn from these data. This thesis seeks to close some apparent gaps between the growing size of emerging datasets and the capabilities of existing approaches to statistical modeling and computing. Specifically, we focus on three problems that arise in learning from high-dimensional data and are of great use in practice. The first problem is to estimate high-dimensional covariance matrix for financial assets via the Barra model, which is one of the most widely used risk models in financial industry. We first study theoretical properties of the Barra model. A surprising conclusion is that as the sample size increases, the Barra approach is in fact not asymptotically consistent. To improve the estimation of the Barra approach, we re-interpret the Barra model via the framework of the random effects model and propose an EM-like method for estimating the Barra model, which is consistent and performs well when the number of assets is large. With the estimated covariance matrix for financial assets, the second problem we investigate is on selecting stable and sparse portfolios. The L1-norm regularized mean-variance portfolio analysis has the advantage of simultaneously controlling the estimation error and performing automatic portfolio selection. We propose an efficient algorithm that combines coordinate descent and augmented Lagrangian methods to solve the optimization problem. To further reduce the computational cost, we also propose a novel screening method for solving the L1-norm regularized optimization problem with an equality constraint. The innovated screening method is able to save substantial computational cost by quickly identifying and removing the assets that are guaranteed to be zero-weighted in the solution. The third problem we consider is to recover the underlying structure of corrupted low rank matrices. Specifically, we assume the observed data matrix is the summation of a low rank matrix, a sparse matrix and noise. We propose a series of spectral regularization algorithms, which are easy to implement and have less computational complexity comparing with existing algorithms. Convergence properties of the proposed algorithms have also been shown under certain conditions.