AI 中文总结
该研究针对统计稀疏学习中高斯分布假设的局限性，提出采用$q$高斯分布，结合改编自流平衡数值方法的框架，开发了用于稀疏统计学习的高效稳健算法，为实际数据分析提供支撑。
AI 中文摘要
本文解决了统计稀疏学习中高斯分布假设的局限性，特别是在对相关且异质的数据进行建模时。传统高斯模型对异常值以及潜在分布假设往往缺乏稳健性。为克服这些局限性，我们提出使用源自Tsallis熵最大化的$q$高斯分布作为一种稳健的替代方案，这在生物统计学中尤为相关，因为相关观测值和异质性（如遗传和纵向研究中的情况）普遍存在。我们的贡献包括通过从Tsallis熵最大化重新推导多元概率密度函数来对相关数据进行建模，从而解决传统高斯模型固有的局限性。此外，我们引入了一种新颖的框架，该框架改编了用于寻找流平衡的数值方法，以解决统计稀疏学习中普遍存在的复合优化问题。将该框架应用于Hager-Zhang共轭梯度算法[Hager2005]，我们开发了一种用于稀疏统计学习的数值稳定且高效的算法。基于Tsallis熵最大化原理的$q$高斯分布，为基于高斯的方法提供了一种可行且灵活的替代方案。本文不仅有助于加深对统计分布和优化技术的理论理解，还为实际数据分析铺平了道路。
英文摘要
This paper addresses the limitations of Gaussian distribution assumptions in statistical sparse learning, particularly in modeling correlated and heterogeneous data. Conventional Gaussian models often lack robustness towards outliers and underlying distribution assumptions. To overcome these limitations, we propose the use of the $q$Gaussian distribution, derived from Tsallis entropy maximization, as a robust alternative. This is notably relevant in biostatistics, where the presence of correlated observations and heterogeneity, such as in genetic and longitudinal studies, is prevalent. Our contributions include modeling of correlated data through the re-derived multivariate probability density function from Tsallis entropy maximization, thereby addressing the limitations inherent in conventional Gaussian models. Furthermore, we introduce a novel framework that adapts numerical methods designed to find equilibria in flows to tackle composite optimization problems prevalent in statistical sparse learning. Applying this framework to the Hager-Zhang conjugate gradient algorithm \cite{Hager2005}, we develop a numerically stable and efficient algorithm for sparse statistical learning. The $q$Gaussian distribution, informed by the principle of maximizing Tsallis entropy, presents a viable and flexible alternative to Gaussian-based methods. This paper not only contributes to the theoretical understanding of statistical distributions and optimization techniques, but also paves the way for practical data analysis.
Comments38 pages; thesis manuscript (July 2024); also available at https://doi.org/10.82308/38780
DOI：10.82308/38780
