Unfolding Generative Flows with Koopman Operators: Fast and Interpretable Sampling

• URL: http://arxiv.org/abs/2506.22304v2
• Date: Tue, 21 Oct 2025 20:27:13 GMT
• Title: Unfolding Generative Flows with Koopman Operators: Fast and Interpretable Sampling
• Authors: Erkan Turan, Aristotelis Siozopoulos, Louis Martinez, Julien Gaubil, Emery Pierson, Maks Ovsjanikov,
• Abstract summary: Continuous Normalizing Flows (CNFs) enable elegant generative modeling but remain bottlenecked by slow sampling.<n>Recent approaches such as Rectified Flow and OT-CFM accelerate sampling by straightening trajectories, yet the learned dynamics remain nonlinear black boxes.<n>We propose globally linearizing flow dynamics via Koopman theory.
• Score: 25.420559502119485
• License: http://creativecommons.org/licenses/by-sa/4.0/
• Abstract: Continuous Normalizing Flows (CNFs) enable elegant generative modeling but remain bottlenecked by slow sampling: producing a single sample requires solving a nonlinear ODE with hundreds of function evaluations. Recent approaches such as Rectified Flow and OT-CFM accelerate sampling by straightening trajectories, yet the learned dynamics remain nonlinear black boxes, limiting both efficiency and interpretability. We propose a fundamentally different perspective: globally linearizing flow dynamics via Koopman theory. By lifting Conditional Flow Matching (CFM) into a higher-dimensional Koopman space, we represent its evolution with a single linear operator. This yields two key benefits. First, sampling becomes one-step and parallelizable, computed in closed form via the matrix exponential. Second, the Koopman operator provides a spectral blueprint of generation, enabling novel interpretability through its eigenvalues and modes. We derive a practical, simulation-free training objective that enforces infinitesimal consistency with the teacher's dynamics and show that this alignment preserves fidelity along the full generative path, distinguishing our method from boundary-only distillation. Empirically, our approach achieves competitive sample quality with dramatic speedups, while uniquely enabling spectral analysis of generative flows.

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