Scalable Minimum-Volume Simplex Estimation with Non-asymptotic Analysis

Abstract: We study the estimation of a $K$-dimensional simplex from $N$ i.i.d.\ points sampled uniformly from its interior; the observations are convex combinations of $K+1$ unknown prototypes. Existing polynomial-time estimators need cubic per-sample work or $O(NK)$ storage and are impractical at $N\sim 10<sup>6$--$10^8$. We propose DeepMVSA, which re-expresses the minimum-volume principle in neural implicit form: a lightweight coordinate network generates the mixing weights and a triangular LU-type parameterization the dual simplex matrix, reducing the trainable-state memory to $O(K<sup>2)$, independent of $N$, and the cost per data pass to $O(NK<sup>2)$. We prove a non-asymptotic sample-complexity bound of the polynomial-time benchmark order for a localized surrogate estimator; an oracle inequality for every global minimizer of the neural objective, with volume-inflation control and an explicit shrinkage bias; a conditional end-to-end error budget separating statistical, approximation, optimization, and enclosure-residual terms on an explicit envelope event; and two-point lower bounds: at any noise level $σ&gt;0$ fixed independently of $N$, the $N<sup>{-1/2}$ scaling is unimprovable in its $N$-exponent. Experiments with up to $N=10<sup>8$ synthetic observations are consistent with the predicted accuracy and scaling, and feasibility on real scenes of $\sim 10<sup>7$ pixels is demonstrated.