Discovering Data Manifold Geometry via Non-Contracting Flows
This addresses the challenge of discovering data manifold geometry for unsupervised learning, offering a scalable method with theoretical guarantees, though it appears incremental as it builds on flow matching and contrasts with isometric approaches.
The paper tackles the problem of constructing a global reference system for unknown data manifolds by learning non-contracting flows that transport samples to a common reference point, resulting in interpretable intrinsic coordinates; empirically, it achieves correct tangent alignment on synthetic manifolds and competitive classification performance on CIFAR-10.
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.