Grouping penalties and its applications to high-dimensional models
2014-06
Loading...
View/Download File
Persistent link to this item
Statistics
View StatisticsJournal Title
Journal ISSN
Volume Title
Title
Grouping penalties and its applications to high-dimensional models
Authors
Published Date
2014-06
Publisher
Type
Thesis or Dissertation
Abstract
Part I: In high-dimensional regression, grouping pursuit and feature selection have their own merits while complementing each other in battling the curse of dimensionality. To seek parsimonious model, we perform simultaneous grouping pursuit and feature selection over an arbitrary undirected graph with each node corresponding to one predictor. When the corresponding nodes are reachable from each other over the graph,regression coefficients can be grouped, whose absolute values are the same or close. This is motivated from gene network analysis, where genes tend to work in groups according to their biological functionalities. Through a nonconvex penalty, we develop a computational strategy and analyze the proposed method. Theoretical analysis indicates that the proposed method reconstructs the oracle estimator, that is, the unbiased least squares estimator given the true grouping, leading to consistent reconstruction of grouping structures and informative features, as well as to optimal parameter estimation. Simulation studies suggest that the method combines the benefit of grouping pursuit with that of feature selection, and compares favorably against its competitors in selection accuracy and predictive performance. An application to eQTL data is used to illustrate the methodology, where a network is incorporated into analysis through an undirected graph.Part II: Gaussian graphical models are useful to analyze and visualize conditional dependence relationships between interacting units. Motivated from network analysis under different experimental conditions, such as gene networks for disparate cancer subtypes, we model structural changes over multiple networks with possible heterogeneities. In particular, we estimate multiple precision matrices describing dependencies among interacting units through maximum penalized likelihood. Of particular interest are homogeneous groups of similar entries across and zero-entries of these matrices, referred to as clustering and sparseness structures, respectively. A non-convex method is proposed to seek a sparse representation for each matrix and identify clusters of the entries across the matrices. Computationally, we develop an efficient method on the basis of difference convex programming, the augmented Lagrangian method and the block-wise coordinate descent method, which is scalable to hundreds of graphs of thousands nodes through a simple necessary and sufficient partition rule, which divides nodes into smaller disjoint subproblems excluding zero-coefficients nodes for arbitrary graphs with convex relaxation. Theoretically, a finite-sample error bound is derived for the proposed method to reconstruct the clustering and sparseness structures. This leads to consistent reconstruction of these two structures simultaneously, permitting the number of unknown parameters to be exponential in the sample size, and yielding the optimal performance of the oracle estimator as if the true structures were given a priori. Simulation studies suggest that the method enjoys the benefit of pursuing these two disparate kinds of structures, and compares favorably against its convex counterpart in the accuracy of structure pursuit and parameter estimation.
Description
University of Minnesota Ph.D. dissertation. June 2014. Major: Statistics. Advisor: Xiaotong Shen. 1 computer file (PDF); xi, 68 pages, appendix A.
Related to
Replaces
License
Collections
Series/Report Number
Funding information
Isbn identifier
Doi identifier
Previously Published Citation
Suggested citation
Zhu, Yunzhang. (2014). Grouping penalties and its applications to high-dimensional models. Retrieved from the University Digital Conservancy, https://hdl.handle.net/11299/165147.
Content distributed via the University Digital Conservancy may be subject to additional license and use restrictions applied by the depositor. By using these files, users agree to the Terms of Use. Materials in the UDC may contain content that is disturbing and/or harmful. For more information, please see our statement on harmful content in digital repositories.