Discovering Data Manifold Geometry via Non-Contracting Flows

TL;DR

This work develops an unsupervised framework for discovering global data-manifold geometry by learning ambient-space vector fields whose non-contracting flows define a global coordinate chart. It treats the problem through a frame-learning lens, enforcing a positive semidefinite Lie derivative of a conformal metric to prevent collapse and using arc-lengths along sequential flows to obtain intrinsic coordinates. Theoretical guarantees show that, on admissible manifolds with a global chart, the learned flows yield a valid global coordinate system, and the method can identify intrinsic dimension via loss behavior when the number of fields is insufficient. Empirically, the approach aligns learned vector fields with manifold tangents on synthetic examples, exposes non-parallelizability on the sphere, and demonstrates scalable, competitive downstream classification on CIFAR-10, indicating promise for geometry-aware representations beyond local embeddings.

Abstract

We introduce an unsupervised approach for constructing a global reference system by learning, in the ambient space, vector fields that span the tangent spaces of an unknown data manifold. In contrast to isometric objectives, which implicitly assume manifold flatness, our method learns tangent vector fields whose flows transport all samples to a common, learnable reference point. The resulting arc-lengths along these flows define interpretable intrinsic coordinates tied to a shared global frame. To prevent degenerate collapse, we enforce a non-shrinking constraint and derive a scalable, integration-free objective inspired by flow matching. Within our theoretical framework, we prove that minimizing the proposed objective recovers a global coordinate chart when one exists. Empirically, we obtain correct tangent alignment and coherent global coordinate structure on synthetic manifolds. We also demonstrate the scalability of our method on CIFAR-10, where the learned coordinates achieve competitive downstream classification performance.

Figures (7)

• Figure 1: (a) A local isometry maps a neighborhood of a surface to its tangent plane while preserving intrinsic distances. On curved surfaces such as a half sphere, this is impossible: Due to curvature, a geodesic neighborhood that is circular on the surface appears as an ellipse in the tangent plane. This illustrates why exact isometric learning cannot be achieved on non-flattenable surfaces. (b) A manifold is parallelizable if it admits a global frame of linearly independent tangent vector fields. The half sphere is shown with smoothly varying tangent directions, illustrating the notion of parallelizability.
• Figure 2: Main Principle of Our Method. Given data to an unknown manifold, our goal is to construct a global reference frame defined by a trainable center $C$. (a) Learning a single unconstrained vector field is insufficient: such a field can always be rescaled to collapse the entire manifold onto $C$. (b) Imposing a non-shrinking constraint—by requiring the Lie derivative of the metric to be positive semi-definite (see Appendix \ref{['app:background']})—prevents this collapse and enforces geometric consistency. (c) By sequentially combining m tangent vector fields (matching the intrinsic dimension of the manifold), we can transport every point to a unique location associated with the global frame’s center while ensuring trajectories remain on the manifold.
• Figure 3: Effect of commuting vector fields on flow simplification When the vector fields commute, the sequential application of flows generated by $(F_1, T_1)$ and $(F_2, T_2)$ is equivalent to integrating a single combined vector field. Our formulation exploits this property to replace a sequence of $n$ ODEs with a single ODE producing the same transport.
• Figure 4: Linear manifold in $\mathbb{R}^4$: Frame learning loss versus number of learned vector fields $m$ for 3-dimensional plane dataset.
• Figure 5: Learning tangent spaces on the sphere (a) The combined vector field $F\,T$. Consistent with the Hairy Ball Theorem, no smooth global tangent vector field exists on the sphere, resulting in singularities near the poles. (b) The two learned tangent vector fields defining local tangent directions.

Theorems & Definitions (17)

• Definition 3.1: Flow
• Definition 3.2: Admissible Manifolds with a Global Chart
• Definition 3.3: Admissible maps induced by flows
• Definition 3.4: Lengths of field line segments
• Theorem 3.1: Loss Minimization when the Dimension is Known
• Theorem 3.2: Main Characterization via Loss Minimization
• Theorem 3.3: Equivalent Loss under Commuting Flows
• Definition 3.1: Tangent and normal decomposition
• Definition 3.2
• Definition 3.3: Projected side lengths