🤖 AI Summary
This work addresses abrupt phenomena in continuous generative models—such as mode locking and semantic collapse during sampling—whose underlying mechanisms remain poorly understood. By modeling the denoising process as a gradient flow on a free energy landscape, the study reveals, for the first time from a differential geometric perspective, that such phase transitions originate from projection caustics on the data support: critical regions where the nearest-point projection ceases to be unique. Building on this insight, the authors propose the Critical Boundary Detector (CBD), which accurately identifies unstable windows along generation trajectories, enabling prediction of mode-commitment moments and targeted intervention in geometrically sensitive regions. The method is validated across toy models, standard diffusion frameworks, and latent text-to-image architectures.
📝 Abstract
Continuous-state generative samplers, including diffusion and flow-matching models, evolve through continuous reverse-time dynamics, yet their samples often undergo abrupt qualitative changes: trajectories commit to modes, semantic alternatives collapse, and small perturbations in narrow time windows can produce large downstream effects. This paper develops a geometric account of such phase-transition-like behaviour. We view denoising as gradient descent on a free energy landscape and show that sharp transitions arise near projection caustics, where the nearest-point projection onto the data support ceases to be unique. Motivated by this perspective, we introduce the Critical Boundary Detector (CBD), as practical diagnostics for score-direction instability. Across toy models, standard diffusion models, and latent text-to-image diffusion models, CBD localises mode commitment, predicts intervention-sensitive windows, and supports targeted control in geometrically sensitive regions. Our results connect geometry of data and dynamics of diffusion generation.