通过几何正则化提升多模态数据的流形学习表现
Manifold Learning with Normalizing Flows: Towards Regularity, Expressivity and Iso-Riemannian Geometry
- 用保距变换优化学习到的黎曼结构
- 平衡映射的平滑性与表达能力，减少建模误差
- 适合需要可解释非线性分析的研究者
现代机器学习越来越依赖高维数据往往位于低维非线性流形附近的假设，即流形假说。通过显式建模数据的几何结构，学习黎曼几何可提升聚类、降维和插值等任务的性能与可解释性。特别是，近年来可学习且可评估的拉回几何已实现规模化，为严谨的非线性数据分析和可解释机器学习开辟了新路径。然而，在真实世界的多模态数据中仍存在畸变与建模误差。本文聚焦于解决多模态场景下的这些问题，提出通过保距化学习到的黎曼结构，并平衡微分同胚参数化的正则性与表达力。在合成数据与真实数据的多个数值实验中验证了该方法的协同有效性。
原文摘要 · Abstract (English)
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.
Thank you to arXiv for use of its open access interoperability. PaperDance 不是 arXiv 官方产品；中文卡片由大模型生成，请以原文为准。