Theory and Methods for High Dimensional Structured Pattern Recovery.
详细信息
- 作者：Rao ; Nikhil Surendra.
- 学历：Doctor
- 年：2014
- 毕业院校：The University of Wisconsin
- Department：Electrical Engineering.
- ISBN：9781321158106
- CBH：3635025
- Country：USA
- 语种：English
- FileSize：1612123
- Pages：191
文摘
In the past few years,there has been an exponential growth in the amount of data collected in many fields. Performing statistical inference on data on a large scale brings with it many challenges. To deal with these challenges,data is typically represented using components that are "simple". These notions of simplicity typically allow tractable methods to be used to perform inference on the data. This thesis focuses on understanding algorithmic,statistical and theoretical questions that arise in large-scale structured model selection problems using big) data. This thesis can be broadly categorized into three parts. The first portion of this thesis involves theoretical contributions for structured sparse signal recovery. The lasso has been a widely used tool to recover signals that are sparse,and the group lasso is a natural extension for signals that exhibit structure amongst the non zero components. However,the group lasso cannot handle the case of overlapping groups efficiently and hence finds limited use. We analyze a method that was proposed to overcome the aforementioned drawback of the group lasso. We derive sample complexity bounds for the group lasso with overlap also called the latent group lasso),and show that the number of measurements needed only depends on the size and number of groups,and not the complexity of overlap between the groups. Furthermore,motivated by applications in functional Magnetic Resonance Imaging and computational biology,we introduce the Sparse Overlapping Sets lasso SOSlasso) that can recover signals that are not only overlapping) group sparse,but many components within a group are also zero. We derive sample complexity bounds for the SOSlasso in linear and logistic regression settings. The SOSlasso generalizes the group lasso with overlap,and can be used to recover structured patterns spanning a wide range of applications. We then turn to algorithms for recovering signals that are simple in a very general sense of the word. The algorithmic framework that we propose can be used for standard sparse recovery,group sparse recovery,low rank matrix completion methods,group sparse regularized problems in multitask learning,among others. The algorithm can also be used to recover signals in cases where no tractable methods exist,such as super resolution applications in signal processing,or cases where existing methods are intractable due to massive memory requirements,such as the group lasso with overlapping groups. The method can also be used to perform regression on large graphs,where the graph can be decomposed into overlapping) edges,cycles and/or cliques. Also,one can use our method to recover signals that are made up of a combination of different structures. Lastly,this thesis focuses on novel applications that involve structured pattern recovery. We show the utility of the SOSlasso on multitask learning in fMRI and gene selection applications in computational biology. We then show a novel modeling scheme for recovering wavelet transform coefficients in inverse problems. Our method to model the coefficients allows us to solve convex recovery problems,while at the same time taking advantage of the structure inherent in the coefficients. We finally conclude the thesis and discuss some ongoing research,and also discuss extensions to the work presented.