Relaxing sub-exponential score error to L2-accurate estimation

Determine whether the sub-exponential tail bound on the score estimation error assumed in Assumption 2 (Sub-exponential score error) can be replaced by the standard L2(q_t)-accurate score estimation assumption across all times t in the analysis of the CollocationDiffusion algorithm for simulating the probability flow ODE under the bounded-plus-noise model (Assumption 1).

Background

The paper establishes high-accuracy, dimension-adaptive sampling guarantees for diffusion models by simulating the probability flow ODE using a collocation-based solver. A key analytical assumption is that the score estimation error has sub-exponential tails uniformly over time, which is stronger than the conventional requirement of L2(q_t)-accurate score estimates commonly used in prior diffusion theory.

The authors explicitly note that relaxing this tail assumption to the standard L2 accuracy would be desirable. Establishing such a relaxation without sacrificing the derived convergence guarantees would align the theory more closely with common assumptions in the diffusion literature while preserving the high-accuracy regime.

References

We leave as open whether this can be relaxed to the more standard assumption of $L_2$-accurate score estimation but note that even under the assumption of perfect score estimation, it was previously not known how to achieve high-accuracy guarantees.

— High-accuracy and dimension-free sampling with diffusions  (2601.10708 - Gatmiry et al., 15 Jan 2026) in Section 2, Diffusion models (Assumption: Sub-exponential score error)