Introduction

Abstract

Applications that require analysis of high-dimensional data have grown significantly during the past decade. In many of these applications, such as bioinformatics, social networking, and mathematical finance, the dimensionality of the data is usually much larger than the number of samples or observations acquired. Therefore statistical inference or data processing would be ill-posed for these underdetermined problems. Fortunately, in some applications the data is known a priori to have an underlying structure that can be exploited to compensate for the deficit of observations. This structure often characterizes the domain of the data by a low-dimensional manifold, e.g. the set of sparse vectors or the set of low-rank matrices, embedded in the high-dimensional ambient space. One of the main goals of high-dimensional data analysis is to design accurate, robust, and computationally efficient algorithms for estimation of these structured data in underdetermined regimes.