流形学习与规范化流：走向正则性、表达性与等距黎曼几何
Manifold Learning with Normalizing Flows: Towards Regularity, Expressivity and Iso-Riemannian Geometry
AI总结：
本文提出通过等距化黎曼结构和平衡正则性与表达性，解决多模态数据中的失真和建模误差问题，提升流形学习的性能和可解释性。
摘要：
现代机器学习越来越多地利用这样的见解：高维数据往往接近低维、非线性流形，这一概念被称为流形假设。通过学习黎曼几何算法显式建模数据的几何结构，可以在聚类、降维和插值等任务中实现性能和可解释性的提升。特别是，学习的拉回几何最近经历了变革性的发展，现在可以扩展到学习和评估，这进一步为系统性的非线性数据分析和可解释的机器学习打开了大门。然而，当考虑现实中的多模态数据时，仍有许多需要改进的地方。本工作聚焦于解决多模态设置中可能出现的失真和建模误差，并提出通过等距化学习的黎曼结构以及平衡微分同胚参数化的正则性和表达性来缓解这两个挑战。我们通过合成和真实数据的多个数值实验展示了所提出方法的协同效应的有效性。
英文摘要：
Modern machine learning increasingly leverages the insight that high-dimensional data often lie near low-dimensional, non-linear manifolds, an idea known as the manifold hypothesis. By explicitly modeling the geometric structure of data through learning Riemannian geometry algorithms can achieve improved performance and interpretability in tasks like clustering, dimensionality reduction, and interpolation. In particular, learned pullback geometry has recently undergone transformative developments that now make it scalable to learn and scalable to evaluate, which further opens the door for principled non-linear data analysis and interpretable machine learning. However, there are still steps to be taken when considering real-world multi-modal data. This work focuses on addressing distortions and modeling errors that can arise in the multi-modal setting and proposes to alleviate both challenges through isometrizing the learned Riemannian structure and balancing regularity and expressivity of the diffeomorphism parametrization. We showcase the effectiveness of the synergy of the proposed approaches in several numerical experiments with both synthetic and real data.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
