Improve the factor 4 in the coverage exponent

Determine whether the intrinsic coverage resolution rate proved for diffusion-model sampling on C^β k-dimensional manifolds can be sharpened by reducing the factor 4 in the exponent δ = Õ(N^{-β/(4k)}), under the same reach and regularity assumptions on the data manifold and without strengthening density assumptions.

Background

The main informal theorem states that, under the manifold hypothesis and coarse score learning, the induced sampling dynamics achieve on-manifold δ-coverage at scale δ = Õ(N^{-β/(4k)}), where β is the manifold smoothness. The authors explicitly note a non-optimized constant in the exponent.

They indicate that improving the constant 4 in the denominator is left for future work, suggesting a potential faster coverage rate under the same assumptions.