Files
Abstract
In this dissertation, we explore the problems of high-dimensional feature screening and sampling techniques using Quantum Walk. In the first project, we introduce a novel feature screening methodology that is robust to the underlying distributions of the data, making it well-suited for high-dimensional heterogeneous data. This method is built upon a dependence measure induced by Wasserstein distance, and Gaussianization of the data. We analyze its non-asymptotic properties. We also establish sure screening and rank consistency properties for the proposed screening method upon mild signal strength conditions. Simulation studies demonstrate that our approach outperforms classical feature screening methods in highly nonlinear and heterogeneous cases. In the second project, we propose a model-free feature screening procedure tailored to high-dimensional quantile regressions. We introduce a novel dependence measure to quantify quantile dependence using Copula theory and corresponding non-parametric Kernel estimator. We derive the optimal bandwidth selection for the estimator, and analyze asymptotic properties of the estimator. We also prove sure screening and rank consistency properties for this screening method upon mild signal strength conditions. Additionally, we propose a data-driven threshold selection method for the screening procedure, which effectively controls false discoveries. The feature screening and FDR control performance of our proposals is validated through simulations. In the third project, we apply our feature screening methods to the U.S. 2020 economic data to identify variables related to the GDP growth rate from 2019 to 2020. Using the selected variables and downstream statistical analysis, we explore strategies for maintaining economic stability during major crises, such as the COVID-19 pandemic. In the final project, we investigate the problem of sampling using the 2-state Quantum Walk on the line. We overview the 2-state Quantum Walk on the line and highlight the limitations of this sampling method. We propose a novel approach that combines the strengths of the 2-state Quantum Walk and Kernel smoothing techniques. Experiments indicate that our proposal outperforms traditional Quantum Walk in terms of both density estimation and sampling efficacy.