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Title: Latent Variable Models: Spectral Methods and Non-convex Optimization
Authors: Wang, Kaizheng
Advisors: Fan, Jianqing
Contributors: Operations Research and Financial Engineering Department
Keywords: clustering
dimension reduction
latent variable models
network analysis
non-convex optimization
spectral methods
Subjects: Statistics
Operations research
Issue Date: 2020
Publisher: Princeton, NJ : Princeton University
Abstract: Latent variable models lay the statistical foundation for data science problems with unstructured, incomplete and heterogeneous information. The significant challenges in computation and memory call for efficient estimation procedures which faithfully output high-quality solutions without much fine-tuning. Comprehensive understanding of them helps build consolidated tool kits for complex problems in the future. This thesis is devoted to developments of principled methods that provably tackle latent variable models of different forms. We first establish theoretical guarantees for several spectral methods that recover latent structures using eigen-decomposition of certain data matrices. For ranking from pairwise comparisons, we investigate a vanilla spectral estimator through $\ell_{\infty}$ perturbation analysis of eigenvectors. This justifies its statistical optimality for identifying the top $K$ items. Next, we develop a general $\ell_p$ theory for PCA in Hilbert spaces with weak signals, obtaining the optimality of spectral clustering in sub-Gaussian mixture models. Based on that, we propose a convenient spectral algorithm for contextual community detection, where one seeks to recover communities in a network given additional node attributes. It is shown to achieve the information threshold for exact recovery. Beyond spectral methods, we develop an optimization-based framework for simultaneous dimension reduction and clustering that seeks to transform the data into a low-dimensional point cloud with well-separated clusters. It is very flexible and handles data that incapacitate many existing procedures. We name the method as Clustering via Uncoupled REgression, or CURE for short. For a linear version of the method under a mixture model, we prove that a perturbed gradient descent algorithm achieves near-optimal statistical precision within reasonable amount of time, even in the absence of good initialization.
Alternate format: The Mudd Manuscript Library retains one bound copy of each dissertation. Search for these copies in the library's main catalog:
Type of Material: Academic dissertations (Ph.D.)
Language: en
Appears in Collections:Operations Research and Financial Engineering

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