UCLA

Many modern statistical problems are inherently nonparametric, including a variety of problems in change point detection, graph models, and robust statistics. Previously, the literature is abundant and well studied in parametric and even linear cases in these problems. And for nonlinear cases, many problems remain open and unresolved. In my dissertation, I would like to bridge this gap by leveraging deep learning techniques to solve these nonparametric statistical problems.

Moreover, with the rapid development of deep learning techniques and machine learning theory, we can now conduct asymptotic analysis in a variety of different neural network spaces. And this allows us to thoroughly analyze the convergence rate of the algorithm using these networks.

My first project proposes an offline methodology in change point detection capable of both regression-type problems and multivariate time-evolving data, and provides rigorous theoretical analysis with guarantees for detecting change points and bounding the distance between true and estimated points, covering both independent and dependent sub-Gaussian data sets.

My second project focuses on the graph model and proposes a novel methodology for modeling the time-evolving data, having the capability to predict both existing points with previous data known and new points with previous data unknown. The proposed model also included an asymptotic analysis of the model convergence rate. Two datasets are used to demonstrate the model performance.

My third project proposes a novel two-step neural network-based model structure combining nuclear norm penalization and output truncation, which is robust to various data contamination mechanisms. The asymptotic analysis on the convergence rate of proposed algorithms is given as well, and the numerical experiments in both synthetic and real datasets are included to demonstrate the model capabilities

In my dissertation, I also include an additional project that I collaborate on with Scientists at the Mayo Clinic. We propose a novel redundancy removal methodology that works on the indexing, archiving, and searching of high-resolution medical images and saves storage and search costs significantly.