Thesis Examination Committee

Prof Kasper Meisner NIELSEN, FINA/HKUST (Chairperson)
Prof Daniel PEREZ PALOMAR, ECE/HKUST (Thesis Supervisor)
Prof Wing-Kin MA, Department of Electronic Engineering, The Chinese University of Hong Kong (External Examiner)
Prof Matthew Robert MCKAY, ECE/HKUST
Prof Roger Shu Kwan CHENG, ECE/HKUST
Prof Bing-Yi JING, MATH/HKUST

Abstract

Sparsity has been successfully applied in almost all the fields of science and engineering, especially in high-dimensional applications where a sparse representation can reduce the complexity of the problem and increase the interpretability of the result. However, requiring a sparse solution translates into a non-convex formulation that, in fact, belongs to the NP-hard class. Therefore, there has been a lot of research on finding approximate formulations that lead into sparse solutions and on deriving efficient algorithms for solving the variants of sparse approximations.
 
In this dissertation we focus on sparsity methods for high-dimensional applications in the fields of machine learning and finance. In particular, we first consider the problem of estimating sparse eigenvectors of a symmetric matrix that attracts a lot of attention in many applications, especially those with a high-dimensional data set. This is the well known principal component analysis (PCA) procedure with a sparsity requirement. Existing methods achieve sparsity at the expense of sacrificing the orthogonality property of the eigenvectors. We develop a new method to estimate sparse eigenvectors without trading off their orthogonality, while outperforming existing algorithms in terms of support recovery and explained variance. Further, we consider the index tracking problem, an application of sparsity coming from the financial industry. Index tracking is a popular passive portfolio management strategy that aims at constructing a portfolio that replicates the performance of an index, e.g., the Hang Send or the S&P 500. The tracking error can be minimized by purchasing all the assets of the index in appropriate amounts. However, to avoid small and illiquid positions and large transaction costs it is desired that the tracking portfolio consists of a small number of assets, i.e., to be sparse. We propose efficient and fast index tracking algorithms under various performance measures, while assuming a set of general portfolio constraints.