Energy Guided Geometric Flow Matching

2025-09-24

AI Summary

High-dimensional time-series generation suffers from trajectory deviation from the underlying data manifold, while conventional flow matching struggles to capture intrinsic geometric structures. Method: We propose energy-guided geometric flow matching, which innovatively integrates score matching with annealed energy distillation to implicitly learn a Riemannian metric tensor reflecting the data’s intrinsic geometry. This enables construction of a nonlinear conditional flow grounded in the learned metric—bypassing dimensionality curse induced by RBF kernels or neighborhood graphs. Unlike existing approaches relying on linear paths or explicit geodesic approximations, our framework adaptively models complex manifold structures. Contribution/Results: Experiments on analytical manifold synthetic data and single-cell trajectory interpolation demonstrate that generated paths better align with true geodesics, yielding significantly improved interpolation accuracy and effectively mitigating manifold deviation in high-dimensional spaces.

Technology Category

• Machine Learning: Learning with Manifolds
• Natural Language Processing: Generation
• Search and Optimization: Learning to Search

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking
• User Modeling, Personalization and Recommendation: Metrics for user behavior and evaluating success

Abstract

A useful inductive bias for temporal data is that trajectories should stay close to the data manifold. Traditional flow matching relies on straight conditional paths, and flow matching methods which learn geodesics rely on RBF kernels or nearest neighbor graphs that suffer from the curse of dimensionality. We propose to use score matching and annealed energy distillation to learn a metric tensor that faithfully captures the underlying data geometry and informs more accurate flows. We demonstrate the efficacy of this strategy on synthetic manifolds with analytic geodesics, and interpolation of cell

Problem

• Learning geodesics without suffering dimensionality curse
• Capturing data geometry via score-based metric tensor
• Improving flow accuracy for temporal data interpolation

Innovation

• Score matching learns metric tensor for data geometry
• Annealed energy distillation informs more accurate flows
• Method avoids dimensionality curse of RBF kernels