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

TL;DR

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.

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.

Figures (13)

• Figure 1: The $\mathbb{R}^2$-valued data in blue, living in a bimodal normal distribution that is visualized by its level sets, reside closely to the manifold in pink (dashed). The geodesic in orange under a data-driven Riemannian structure (see \ref{['app:illustrative-pullback']}) follows the manifold (and the regions of high likelihood) in the underlying data distribution in a natural fashion.
• Figure 2: Both geodesic interpolation and non-linear rank-1 approximation under a pullback structure on $\mathbb{R}^2$ used in Figure \ref{['fig:double-gaussian-data-analysis-results-toy-metric-geo']} suffer from distortions that are detrimental for interpolate interpretability and rank-1 reconstruction performance.
• Figure 3: Neither isometrized geodesic interpolation nor isometrized non-linear rank-1 approximation under a pullback structure on $\mathbb{R}^2$ suffer from distortions that the initial pullback structure suffered from (see Figure \ref{['fig:double-gaussian-data-analysis-results-affine-unbend-metric']}).
• Figure 4: Both geodesic interpolation and non-linear rank-1 approximation under a learned pullback structure Diepeveen2024a on $\mathbb{R}^2$ do not behave as expected, which is detrimental for interpolate interpretability and rank-1 reconstruction fairness.
• Figure 5: Neither geodesic interpolation nor non-linear rank-1 approximation under a more regular learned pullback structure on $\mathbb{R}^2$ have learned an incorrect geometry, in contrast to the initial pullback structure (see Figure \ref{['fig:double-gaussian-data-analysis-results-affine-metric']}).

Theorems & Definitions (26)

• Remark 1
• Theorem 1
• Theorem 2
• Theorem 3
• Theorem 4
• Theorem 5
• Remark 2
• Theorem 6
• Theorem 7
• Remark 3