Methods for High Dimensional Inferences With Applications in Genomics

In this dissertation, I have developed several high dimensional inferences and computational methods motivated by problems in genomics studies. It consists of two parts. The first part is motivated by analysis of data from genome-wide association studies (GWAS), where I have developed an optimal false discovery rate (FDR) con- trolling method for high dimensional dependent data. For short-ranged dependent data, I have shown that the marginal plug-in procedure has the optimal property in controlling the FDR and minimizing the false non-discovery rate (FNR). When applied to analysis of the neuroblastoma GWAS data, this procedure identified six more disease-associated variants compared to previous p-value based procedures such as the Benjamini and Hochberg procedure. I have further investigated the statistical issue of sparse signal recovery in the setting of GWAS and developed a rigorous procedure for sample size and power analysis in the framework of FDR and FNR for GWAS. In addition, I have characterized the almost complete discovery boundary in terms of signal strength and non-null proportion and developed a procedure to achieve the almost complete recovery of the signals. The second part of my dissertation was motivated by gene regulation network construction based on the genetical genomics data (eQTL). I have developed a sparse high dimensional multivariate regression model for studying the conditional independent relationships among a set of genes adjusting for possible genetic effects, as well as the genetic architecture that influences the gene expression. I have developed a covariate adjusted precision matrix estimation method (CAPME), which can be easily implemented by linear programming. Asymptotic convergence rates and sign consistency are established for the estimators of the regression coefficients and the precision matrix. Numerical performance of the estimator was investigated using both simulated and real data sets. Simulation results have shown that the CAPME resulted in great improvements in both estimation and graph structure selection. I have applied the CAPME to analysis of a yeast eQTL data in order to identify the gene regulatory network among a set of genes in the MAPK signaling pathway. Finally, I have also made the R software package CAPME based on my dissertation work

First-order Convex Optimization Methods for Signal and Image Processing

In this thesis we investigate the use of first-order convex optimization methods applied to problems in signal and image processing. First we make a general introduction to convex optimization, first-order methods and their iteration com-plexity. Then we look at different techniques, which can be used with first-order methods such as smoothing, Lagrange multipliers and proximal gradient meth-ods. We continue by presenting different applications of convex optimization and notable convex formulations with an emphasis on inverse problems and sparse signal processing. We also describe the multiple-description problem. We finally present the contributions of the thesis. The remaining parts of the thesis consist of five research papers. The first paper addresses non-smooth first-order convex optimization and the trade-off between accuracy and smoothness of the approximating smooth function. The second and third papers concern discrete linear inverse problems and reliable numerical reconstruction software. The last two papers present a convex opti-mization formulation of the multiple-description problem and a method to solve it in the case of large-scale instances. i i