Topics in High-dimensional and Large-scale Data Analysis

This thesis concerns the analysis of high-dimensional and large-scale data that have become ubiq-uitous in today’s information-driven age. It consists of four main chapters. The first studies the problem of variable selection, where out of potentially thousands of measured variables, one wishes to select just a few that are relevant for a particular phenomenon of interest. Here, we develop further methodology and theory for Stability Selection, an important variable selection technique introduced in Meinshausen and Bühlmann (2010) that provides an upper bound on the expected number of irrelevant variables selected. Unfortunately the bound requires a strong exchangeability condition and it can be rather weak at times. We introduce a version of the bound that does not require exchangeability assumptions, and in addition, under some mild conditions, we obtain tighter bounds that can be used by practitioners to yield more true discoveries for the same level of error tolerance. In Chapter 2 we consider the problem of high-dimensional regression when there may be in-teracting variables. We introduce a new method that searches for interactions in a hierarchical fashion. The procedure is very general: it can be incorporated into many high-dimensional regres-sion algorithms for fitting only additive models. It is computationally fast, making use of paralle