Scalable Minimum-Volume Simplex Estimation with Non-asymptotic Analysis

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~ 106--108. 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(K2), independent of N, and the cost per data pass to O(NK2). 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>0 fixed independently of N, the N-1/2 scaling is unimprovable in its N-exponent. Experiments with up to N=108 synthetic observations are consistent with the predicted accuracy and scaling, and feasibility on real scenes of ~ 107 pixels is demonstrated.