Keywords

Low-rank matrix, stochastic block model, clustering, generalized linear mode, ill-conditioned matrix recovery

Abstract

The first part of this dissertation focuses on matrix-covariate regression models. While they have been studied in many existing works, classical statistical and computational methods for the analysis of the regression coefficient estimation are highly affected by high dimensional matrix-valued covariates. To address these issues, we proposes a framework of matrix-covariate regression models based on a low-rank constraint and an additional regularization for structured signals, with considerations of models of both continuous and binary responses. In the second part, we examine a Mixture Multilayer Stochastic Block Model (MMLSBM), where layers can be grouped into sets of similar networks. Each group of networks is endowed with a unique Stochastic Block Model. The objective is to partition the multilayer network into clusters of similar layers and identify communities within those layers. We present an alternative approach called the Alternating Minimization Algorithm (ALMA), which aims to simultaneously recover the layer partition and estimate the matrices of connection probabilities for the distinct layers. In the last part, we demonstrates the effectiveness of the projected gradient descent algorithm. Firstly, its local convergence rate is independent of the condition number. Secondly, under conditions where the objective function is rank-2r restricted L-smooth and μ-strongly convex, with L/μ < 3, projected gradient descent with appropriate step size converges linearly to the solution. Moreover, a perturbed version of this algorithm effectively navigates away from saddle points, converging to an approximate solution or a second-order local minimizer across a wide range of step sizes. Furthermore, we establish that there are no spurious local minimizes in estimating asymmetric low-rank matrices when the objective function satisfies L/μ < 3.

Completion Date

2024

Semester

Spring

Committee Chair

Zhang, Teng

Degree

Doctor of Philosophy (Ph.D.)

College

College of Sciences

Department

Mathematics

Format

application/pdf

Language

English

Rights

In copyright

Release Date

May 2027

Length of Campus-only Access

3 years

Access Status

Doctoral Dissertation (Campus-only Access)

Campus Location

Orlando (Main) Campus

Accessibility Status

Meets minimum standards for ETDs/HUTs

Restricted to the UCF community until May 2027; it will then be open access.

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