Polynomial-time attainment of the noiseless minimax rate

Determine whether any polynomial-time simplex estimator can attain the noiseless minimax sample-complexity rate \(\Theta(K/N)\) under the uniform simplex-sampling model.

Background

The paper compares polynomial-time procedures with the information-theoretically optimal but NP-hard maximum-likelihood estimator. The latter achieves the noiseless minimax order Θ(K/N)\Theta(K/N), whereas the polynomial-time benchmark analyzed in the paper has an N−1/2N^{-1/2}-type statistical rate. The authors explicitly leave unresolved whether a polynomial-time method can achieve the faster information-theoretic rate.

References

In this paper we propose Deep Minimum Volume Simplex Analysis (DeepMVSA), a neural implicit simplex estimator whose memory footprint and parameter complexity are independent of $N$, and we answer all three questions for it; whether any polynomial-time estimator can attain the noiseless minimax rate $\Theta(K/N)$ remains open.

— Scalable Minimum-Volume Simplex Estimation with Non-asymptotic Analysis  (2609.25576 - Li et al., 22 Sep 2026) in Section 1, Introduction

The experiments therefore probe the mechanisms identified by the theory---the approximation floor of Remark~\ref{rem:apxfloor}, the capacity prescription of Proposition~\ref{prop:apxrate}, and the scaling laws of Theorem~\ref{thm:complexity}---rather than verifying the theorem's constants; narrowing the gap between the combinatorial threshold, inherited from the window argument of Lemma~S2 of , and practical sample sizes remains open.

— Scalable Minimum-Volume Simplex Estimation with Non-asymptotic Analysis  (2609.25576 - Li et al., 22 Sep 2026) in Theorem 2, Sample complexity of the surrogate DeepMVSA estimator