AI 中文总结
本文改进了切片正态（SN）分布框架，将其参数估计转化为凸优化问题，明确其表达能力并提出高维拟合流程，在NASA失速飞行数据上验证了该方法可捕捉非线性依赖模式。
AI 中文摘要
为下游应用构建兼具非线性依赖、多模态特性与可处理解析结构的多元分布模型，是不确定性量化领域的核心挑战。美国国家航空航天局（NASA）的前期研究中提出了切片正态（Sliced Normal, SN）分布，该分布通过多项式特征映射表示密度，为更不透明的生成模型提供了紧凑的代数替代方案，同时保留了捕捉非线性参数依赖与多模态行为的能力。本文在SN框架基础上进行改进，提升了方法的可靠性与可扩展性：首先，将SN参数估计重新表述为半正定矩阵上的凸优化问题，以适用于标准优化工具的形式替代了原始非凸似然搜索；其次，通过将多项式对数密度建模与紧域上的Stone–Weierstrass型通用近似论证相联系，明确了SN类的表达能力；最后，提出了一种高维拟合流程，将变量划分为近似独立的组，在每个子组内拟合SN模型，再通过跨块补全步骤组装子组模型以恢复残差依赖。我们将该SN建模流程应用于NASA失速飞行数据，结果表明该方法可捕捉低维切片及高维块组装模型中的非线性依赖模式。
英文摘要
Modeling multivariate distributions with nonlinear dependence, multimodality, and tractable analytical structure for downstream applications is a central challenge in uncertainty quantification. Sliced Normal (SN) distributions were introduced in prior works at the National Aeronautics and Space Administration (NASA) to address this need by representing densities through polynomial feature maps. This construction provides a compact algebraic alternative to more opaque generative models, while retaining the ability to capture nonlinear parameter dependencies and multi-modal behavior. In this paper, we build on the SN framework and develop several improvements that make the approach more reliable and scalable. First, we reformulate SN parameter estimation as a convex optimization problem over a positive semidefinite matrix, replacing the original nonconvex likelihood search with a formulation amenable to standard optimization tools. Second, we clarify the expressive power of the SN class by connecting polynomial log-density modeling to a Stone--Weierstrass-type universal approximation argument on compact domains. Third, we propose a high-dimensional fitting procedure that partitions variables into approximately independent groups, fits SN models within each subgroup, and then assembles the subgroup models through a cross-block completion step to recover residual dependence. We demonstrate the resulting SN modeling pipeline on NASA loss-of-control flight data, where the method captures nonlinear dependence patterns in both low-dimensional slices and a higher-dimensional block-assembled model.
