Projection Bottleneck in Adaptive Diffusion Sampling

Adaptive timestep selection is widely used to accelerate sampling in continuous-time generative models, yet its effectiveness remains poorly understood. Existing methods generally posit that structured trajectory signals can be leveraged to derive improved discretization schemes. In this work, we show that this assumption can break down for projection-based step-wise adaptation under fixed sampling budgets. We find that trajectory-level signals, including hidden speed, curvature, and topological summaries, exhibit strong and consistent temporal structure across models and datasets. However, directly using these signals for step-wise adaptive control does not consistently improve sampling quality and can lead to severe degradation. Empirically, diverse controllers often produce identical timestep schedules and even identical generated samples. We identify an important mechanism underlying this behavior as a projection bottleneck: under discretization constraints, distinct adaptive proposals can be mapped onto the same feasible schedule through a many-to-one projection. We characterize this effect theoretically through active-set non-injectivity and local loss of controller degrees of freedom under saturation. Our results suggest that effective adaptive sampling depends not only on extracting trajectory information, but also on preserving that information during schedule construction, motivating alternative allocation strategies that avoid projection-induced information loss.