Beyond Unimodal Bases: Pullback Geometry for Multimodal Data

2026-09-30

AI Summary

This study addresses the transport deformation and geometric distortion arising in multimodal data when existing unimodal Gaussian manifold geometries enforce restrictive mappings. To overcome the single-manifold assumption, this work proposes a pullback geometry based on latent Gaussian mixtures, defining a Riemannian structure via responsibility-weighted precision metrics to establish a smooth positive-definite metric over mixture manifolds. It further proves that geodesics exhibit concavity along the log-density, ensuring paths traverse high-likelihood regions. The proposed framework integrates adaptive mixture learning with normalizing flows and local effective dimensionality estimation for robust modeling. Evaluations on synthetic and MNIST datasets demonstrate that this approach significantly reduces transport distortion, enhances interpolation realism and reference trajectory recovery accuracy, while preserving the interpretability of local structures.

Abstract

Data-driven Riemannian geometry provides nonlinear interpolation and geometric representations of high-dimensional data. For these operations to be statistically meaningful, paths between observations should preferentially traverse high-likelihood regions. Existing scalable pullback constructions typically use a unimodal Gaussian latent distribution, assuming that the data reside close to a single manifold. For multimodal data, mapping separated modes or local structures into one Gaussian region can require substantial transport deformation and compromise the resulting geometry. We introduce a pullback geometry for data supported on mixtures of manifolds. Using a latent Gaussian mixture, we define its Riemannian metric as the matrix square of the responsibility-weighted expected component precision. The metric is smooth and positive definite and recovers the existing Gaussian construction in the single-component limit. For structured overlapping mixtures, we establish conditions under which the log-density is concave along geodesics, providing a formal connection between the proposed geometry and paths through high-likelihood regions, and derive the corresponding local curvature relations. We instantiate this geometry in a normalizing flow with adaptive mixture learning, allowing the number of active components to emerge from the data and supporting component-wise reconstruction and local effective-dimension estimation. Experiments on synthetic geometric data, a controlled multi-view image setting with a known reference trajectory, and MNIST show reduced transport distortion, competitive path support, close reference-trajectory recovery, and improved interpolation realism. These results extend scalable pullback geometry beyond datasets that reside close to a single manifold while retaining tractable and interpretable local structure.

Problem

• pullback geometry
• multimodal data
• Riemannian geometry
• mixture of manifolds
• transport deformation

Innovation

• Pullback Geometry
• Riemannian Metric
• Gaussian Mixture Model
• Normalizing Flow
• Multimodal Data