Discovering Data Manifold Geometry via Non-Contracting Flows

2026-02-02

AI Summary

This work addresses the problem of constructing a global intrinsic coordinate system and learning the geometric structure of an unknown data manifold under unsupervised conditions. To this end, the authors propose a non-collapsing flow approach that learns vector fields in the ambient space spanning the manifold’s tangent spaces and transports all samples along these flows toward a learnable common reference point, thereby establishing a consistent global coordinate chart. The method incorporates a non-contraction constraint and a flow-matching objective that avoids explicit integration, theoretically enabling recovery of a globally consistent coordinate map while preventing manifold collapse and relaxing the restrictive isometric assumptions common in prior work. Experiments demonstrate that the approach accurately aligns tangent spaces and constructs coherent coordinates on synthetic manifolds, and achieves competitive downstream classification performance on CIFAR-10.

Technology Category

• Machine Learning: Learning with Manifolds
• Computer Vision: Learning & Optimization for CV
• Search and Optimization: Non-convex Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Economics, Online Markets and Human Computation: Data quality aspects of human-annotated datasets
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking

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.

Problem

• data manifold
• global reference system
• intrinsic coordinates
• tangent vector fields
• non-contracting flows

Innovation

• non-contracting flows
• tangent vector fields
• global coordinate chart
• manifold learning
• flow matching