Manifold Learning with Normalizing Flows: Towards Regularity, Expressivity and Iso-Riemannian Geometry

2025-05-12

AI Summary

To address metric distortion in pullback Riemannian geometry and diffeomorphic modeling error in multimodal data, this paper proposes the Iso-Riemannian manifold learning framework. It introduces isometric constraints into normalized-flow-driven pullback geometric learning for the first time, jointly optimizing diffeomorphic mappings and their induced pullback metrics. By incorporating Jacobian-based geometric regularization and differentiable manifold optimization, the method explicitly balances mapping regularity and representational capacity. Evaluated on synthetic and real-world multimodal datasets, it significantly improves interpolation continuity, clustering consistency, and distance preservation in low-dimensional representations—reducing geometric error by 37% and enhancing interpretability of downstream tasks. The core contribution is the establishment of the first scalable and interpretable isometric Riemannian manifold learning paradigm.

Technology Category

• Machine Learning: Learning with Manifolds
• Intelligent Robots: Multimodal Perception & Sensor Fusion
• Computer Vision: Representation Learning for Vision

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Search and Retrieval-Augmented AI: Retrieval-Augmented Generation (RAG) and multi-modal RAG
• User Modeling, Personalization and Recommendation: Fairness-aware retrieval and ranking

Abstract

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.

Problem

• Address distortions in multi-modal data manifold learning
• Isometrize learned Riemannian structure for better modeling
• Balance regularity and expressivity in diffeomorphism parametrization

Innovation

• Isometrizing learned Riemannian structure
• Balancing regularity and expressivity
• Using normalizing flows for manifold learning