Beyond Unimodal Bases: Pullback Geometry for Multimodal Data

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.