Manifold Learning with Normalizing Flows: Towards Regularity, Expressivity and Iso-Riemannian Geometry
This work addresses challenges in manifold learning for real-world multi-modal data, representing an incremental improvement in geometric machine learning.
The paper tackles distortions and modeling errors in multi-modal data by isometrizing learned Riemannian structures and balancing regularity and expressivity in diffeomorphism parametrizations, showing effectiveness in synthetic and real data experiments.
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.