Joint minimax rate in the noisy simplex model

Characterize the noisy-model minimax estimation rate as a joint function of the sample size \(N\) and Gaussian noise level \(\sigma\) for uniform simplex sampling.

Background

The paper proves a lower bound showing that, for fixed positive noise, every estimator has an N−1/2N^{-1/2} error exponent up to geometric and noise factors. It also discusses a noise-induced accuracy floor associated with the surrogate analysis. However, these results do not determine the complete minimax dependence on both NN and σ\sigma, particularly across regimes in which the noise level varies with the sample size. The authors explicitly state that this joint noisy-model rate remains unresolved.

References

Part (i) certifies only this exponent: at fixed $\sigma$ the noise-induced floor of Proposition~\ref{prop:noise} is bounded away from zero while $\epsilon_\sigma\to0$, so the noisy-model minimax rate as a joint function of $(N,\sigma)$ remains open.

— Scalable Minimum-Volume Simplex Estimation with Non-asymptotic Analysis  (2609.25576 - Li et al., 22 Sep 2026) in Section 2, Proposition 5 discussion: Lower bounds and necessity of the gap cap