Optimize constants and identify sharp minimax dependencies

Identify sharp constants and precise minimax parameter dependencies in the bounds for manifold recovery, projection accuracy, and coverage, and tighten the current results stated up to polylogarithmic and geometry- or density-dependent constants.

Background

The established rates and guarantees are given up to unspecified polylogarithmic factors and depend on manifold and density parameters such as reach and density bounds.

The authors explicitly leave the tightening of these constants and the determination of sharp minimax dependencies to future work.

References

We have made no attempt to optimize constants: bounds are stated up to polylogarithmic factors and manifold- and density-dependent constants (e.g., reach and density bounds). Tightening these constants and identifying sharp minimax dependencies is left for future work.

— Manifold Generalization Provably Proceeds Memorization in Diffusion Models  (2603.23792 - Shen et al., 24 Mar 2026) in Conclusion, Open directions (4)

Although the accuracy of this method depends on the number of uniform samples on the unit ball surface, the precise relationship between the local projection error and the sample count remains an open question.

— On Solving Continuous-Discrete Projection Filters via Sum-of-Squares Relaxation  (2608.28002 - Emzir, 28 Aug 2026) in Section I, Introduction