Scalable Minimum-Volume Simplex Estimation with Non-asymptotic Analysis

arXiv:2609.25576v1 Announce Type: cross 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 Nsim 10^6--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^2), independent of N, and the cost per data pass to O(NK^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 sigma>0 fixed independently of N, the N^{-1/2} scaling is unimprovable in its N-exponent. Experiments with up to N=10^8 synthetic observations are consistent with the predicted accuracy and scaling, and feasibility on real scenes of sim 10^7 pixels is demonstrated.

Source: arXiv cs.LG | 2026-09-23