The Geometry of Phase Transitions in Generative Dynamics via Projection Caustics
For researchers in generative modeling, this work offers a geometric framework to understand and control abrupt qualitative changes in diffusion dynamics, though it is an incremental theoretical contribution with practical diagnostics.
This paper provides a geometric explanation for phase transitions in diffusion and flow-matching models, showing that sharp transitions occur near projection caustics where nearest-point projection onto data support is non-unique. They introduce the Critical Boundary Detector (CBD) to localize mode commitment and predict intervention-sensitive windows, validated across toy models and standard diffusion models.
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.