Manifold-Stable Flow Matching

2026-09-28

AI Summary

This study addresses the challenge in flow matching where generated samples struggle to remain on the data manifold without geometric priors. To this end, we propose Manifold-Stabilized Flow Matching (MSFM), a framework integrating tangential learning with normal contraction theory. By leveraging probability path decomposition, PCA-based local affine surrogates, and an analytical projector, MSFM achieves manifold invariance and controllable time-accurate convergence from arbitrary prior distributions. As the first contraction-theoretic manifold-stabilized framework, it ensures geometric consistency without requiring priors predefined on the manifold support. Experimental results demonstrate that ellipse fitting errors and rotational deviations are reduced to the orders of 10⁻⁶ and 10⁻⁷, respectively, while success rates in robotic tasks improve significantly, validating the superior geometric adherence of the proposed method.

Abstract

Flow matching (FM) learns generative dynamics through velocity regression. Geometric FM variants commonly assume a prior supported on the data manifold, requiring geometric knowledge that is often unavailable. Without such knowledge, low regression error alone does not guarantee manifold adherence. Adherence keeps generated samples within valid configurations and is empirically associated with better task performance. We introduce manifold-stable flow matching (MSFM), which can start from an arbitrary ambient prior, not necessarily supported on the manifold. Using tools from nonlinear dynamics, namely contraction theory, MSFM combines learned tangential transport with prescribed normal contraction. The construction uses analytical projectors for known manifolds and local affine proxies estimated by principal component analysis for unknown data geometry. By implementing contraction theory in both cases of known and unknown manifolds, we guarantee manifold invariance and transverse convergence to the manifold within a desired time window (e.g., one second). We derive a family of compatible probability paths and decompose the training loss into a learnable tangential term and a normal residual. An ellipse experiment attains a mean terminal off-manifold error of order $10^{-6}$. In Push-T robotic experiments, MSFM raises success from $74\%$ to $82\%$. In the Robomimic Square task, success increases from $60\%$ to $72\%$, while rotation-manifold deviation decreases from order $10^{-2}$ to $10^{-7}$. The MSFM terminal geometric errors are controlled by the chosen numerical tolerance. These results demonstrate stronger geometric adherence and higher observed task performance, supporting prescribed normal contraction as a complement to learned generative transport.

Problem

• Flow Matching
• Manifold Adherence
• Generative Models
• Geometric Constraints

Innovation

• Flow Matching
• Contraction Theory
• Manifold Stability
• Generative Models
• Principal Component Analysis