Title: On the Collapse of Generative Paths: A Criterion and Correction for Diffusion Steering
Primary area: generative models
Keywords: Diffusion Model, Flow Matching, Inference-time Steering, Feynman-Kac, Probability Path, Scaffold Decoration, Compositional Image Generation

Abstract: Inference-time steering is vital for adapting large-scale diffusion models to scientific and creative tasks. A powerful and widely-used steering paradigm is the ratio-of-densities, which reweights the data landscape to amplify desired attributes, such as in classifier-free guidance. However, this method harbors a critical, previously unformalized failure mode: Marginal Path Collapse. We show that even when the start and end points of the generative process are valid distributions, the path between them can break down, causing the intermediate densities to become mathematically ill-defined. This is not a rare edge case but a systematic flaw affecting most popular noise schedules, especially when composing heterogeneous models.
    In this work, we provide a complete solution. First, we establish a rigorous path existence criterion, a simple diagnostic test that precisely predicts the onset of collapse. Second, based on this criterion, we develop Adaptive path Correction with Exponents (ACE), a principled framework that provably prevents collapse. By extending the Feynman-Kac partial differential equation for generative control to support time-varying exponents, ACE dynamically adjusts the steering weights to guarantee a valid probability path from noise to data. We validate our framework on synthetic benchmarks, where ACE eliminates collapse and reduces distributional error by over 5x compared to prior methods. We then demonstrate its practical necessity in the high-stakes application of molecular design, where ACE enables stable scaffold-based optimization that is unattainable with existing techniques. Our work transforms ratio-of-densities steering from an unstable heuristic into a robust and theoretically-grounded tool for controllable generation.

TL;DR: Ratio-of-densities steering in diffusion models breaks via Marginal Path Collapse. We introduce ACE, a provably stable correction that prevents collapse, cuts error >5×, and enables reliable molecular design.