Discovering Data Manifold Geometry via Non-Contracting Flows

TL;DR

This paper presents a novel unsupervised method for learning a global coordinate system on data manifolds by training vector fields that transport data points to a common reference, enabling interpretable intrinsic coordinates.

Contribution

It introduces a non-contracting flow-based approach that learns tangent vector fields without assuming flatness, with theoretical guarantees and scalable empirical results.

Findings

Successfully recovers global coordinates on synthetic data

Achieves coherent tangent alignment and interpretable coordinates

Demonstrates competitive classification performance on CIFAR-10

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.