Unfolding Generative Flows with Koopman Operators: Fast and Interpretable Sampling

2025-06-27

AI Summary

Conditional flow matching (CFM) suffers from computationally expensive nonlinear ODE solvers and limited interpretability. To address these limitations, this paper proposes Koopman-CFM—a decoder-agnostic generative modeling framework. Its core innovation lies in leveraging Koopman operator theory to embed nonlinear generative flows into a linear observable space, enabling closed-form linear dynamical modeling of generation trajectories. This formulation permits single-step sampling via matrix exponentiation and provides spectral analysis tools to characterize temporal scales and stability properties of the generative process. Crucially, Koopman-CFM operates without diffusion assumptions. Empirical evaluation on MNIST, Fashion-MNIST, and the Textured-Face Dataset demonstrates substantial acceleration in sampling speed while preserving both computational efficiency and structural interpretability.

Technology Category

• Machine Learning: Deep Generative Models & Autoencoders
• Natural Language Processing: Generation
• Computer Vision: Diffusion Models for Vision

Application Category

• Economics, Online Markets and Human Computation: Economic ramifications for generative AI infrastructure and applications
• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Social Networks and Social Media: Generative AI / large language models and their impact on social systems

Abstract

Conditional Flow Matching (CFM) offers a simulation-free framework for training continuous-time generative models, bridging diffusion and flow-based approaches. However, sampling from CFM still relies on numerically solving non-linear ODEs which can be computationally expensive and difficult to interpret. Recent alternatives address sampling speed via trajectory straightening, mini-batch coupling or distillation. However, these methods typically do not shed light on the underlying extit{structure} of the generative process. In this work, we propose to accelerate CFM and introduce an interpretable representation of its dynamics by integrating Koopman operator theory, which models non-linear flows as linear evolution in a learned space of observables. We introduce a decoder-free Koopman-CFM architecture that learns an embedding where the generative dynamics become linear, enabling closed-form, one-step sampling via matrix exponentiation. This results in significant speedups over traditional CFM as demonstrated on controlled 2D datasets and real-world benchmarks, MNIST, Fashion-MNIST (F-MNIST), and the Toronto Face Dataset (TFD). Unlike previous methods, our approach leads to a well-structured Koopman generator, whose spectral properties, eigenvalues, and eigenfunctions offer principled tools for analyzing generative behavior such as temporal scaling, mode stability, and decomposition in Koopman latent space. By combining sampling efficiency with analytical structure, Koopman-enhanced flow matching offers a potential step toward fast and interpretable generative modeling.

Problem

• Accelerate sampling in Conditional Flow Matching models
• Improve interpretability of generative dynamics structure
• Enable closed-form one-step sampling via Koopman theory

Innovation

• Uses Koopman operator for linear generative dynamics
• Enables closed-form one-step sampling via matrix exponentiation
• Provides interpretable spectral analysis of generative behavior