标题： 具有机器学习的流形分析发现 显示英文标题

标题： Analytical Discovery of Manifold with Machine Learning

摘要： 理解高维数据中的低维结构对于复杂数据集的可视化、解释和去噪至关重要。尽管流形学习技术取得了进展，但一些关键挑战——如全局洞察力有限和缺乏可解释的分析描述——仍未得到解决。 在这项工作中，我们引入了一个新的框架，即使用自编码的全局解析流形学习（GAMLA）。GAMLA 在自编码框架内采用两轮训练过程，以推导出底层流形的特征表示和互补表示。通过特征表示，流形由一个参数化函数表示，该函数展开流形以提供全局坐标。而通过互补表示，则开发了近似的显式流形描述，为高维数据集下的平滑流形提供了全局和解析的表示。这使得可以解析推导出曲率和法向量等几何属性。此外，我们发现这两种表示一起分解了整个潜在空间，并且因此能够表征围绕流形的局部空间结构，在异常检测和分类中表现出特别的有效性。通过在基准数据集和现实应用上的大量实验，GAMLA 展现了其在实现计算效率和可解释性的同时提供精确几何和结构洞见的能力。这一框架弥合了数据驱动的流形学习与解析几何之间的差距，为探索复杂数据集的内在特性提供了一种多功能工具。 显示英文摘要

摘要： Understanding low-dimensional structures within high-dimensional data is crucial for visualization, interpretation, and denoising in complex datasets. Despite the advancements in manifold learning techniques, key challenges-such as limited global insight and the lack of interpretable analytical descriptions-remain unresolved. In this work, we introduce a novel framework, GAMLA (Global Analytical Manifold Learning using Auto-encoding). GAMLA employs a two-round training process within an auto-encoding framework to derive both character and complementary representations for the underlying manifold. With the character representation, the manifold is represented by a parametric function which unfold the manifold to provide a global coordinate. While with the complementary representation, an approximate explicit manifold description is developed, offering a global and analytical representation of smooth manifolds underlying high-dimensional datasets. This enables the analytical derivation of geometric properties such as curvature and normal vectors. Moreover, we find the two representations together decompose the whole latent space and can thus characterize the local spatial structure surrounding the manifold, proving particularly effective in anomaly detection and categorization. Through extensive experiments on benchmark datasets and real-world applications, GAMLA demonstrates its ability to achieve computational efficiency and interpretability while providing precise geometric and structural insights. This framework bridges the gap between data-driven manifold learning and analytical geometry, presenting a versatile tool for exploring the intrinsic properties of complex data sets.