# Scalable Minimum-Volume Simplex Estimation with Non-asymptotic Analysis

arXiv: https://arxiv.org/abs/2609.25576 · HTML: https://mlsanity.com/paper/2609.25576/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 17 of its 42 references that are in the map.
- First posted on arXiv 22 Sep 2026.

## Builds on (most cited first, 17 of 17)

- [Implicit Neural Representations with Periodic Activation Functions](https://mlsanity.com/paper/2006.09661/) (2020)
- [Fourier Features Let Networks Learn High Frequency Functions in Low Dimensional Domains](https://mlsanity.com/paper/2006.10739/) (2020)
- [Density estimation using Real NVP](https://mlsanity.com/paper/1605.08803/) (2016)
- [Glow: Generative Flow with Invertible 1x1 Convolutions](https://mlsanity.com/paper/1807.03039/) (2018)
- [Nonnegative Matrix Factorization for Signal and Data Analytics: Identifiability, Algorithms, and Applications](https://mlsanity.com/paper/1803.01257/) (2018)
- [The Why and How of Nonnegative Matrix Factorization](https://mlsanity.com/paper/1401.5226/) (2014)
- [Endmember-Guided Unmixing Network (EGU-Net): A General Deep Learning Framework for Self-Supervised Hyperspectral Unmixing](https://mlsanity.com/paper/2105.10194/) (2021)
- [Deep Hyperspectral Unmixing using Transformer Network](https://mlsanity.com/paper/2203.17076/) (2022)
- [Maximum Volume Inscribed Ellipsoid: A New Simplex-Structured Matrix Factorization Framework via Facet Enumeration and Convex Optimization](https://mlsanity.com/paper/1708.02883/) (2017)
- [Non-Asymptotic Performance Guarantees for Neural Estimation of mathsff-Divergences](https://mlsanity.com/paper/2103.06923/) (2021)
- [Using SVD for Topic Modeling](https://mlsanity.com/paper/1704.07016/) (2017)
- [Max-Sliced Mutual Information](https://mlsanity.com/paper/2309.16200/) (2023)
- [Probabilistic Simplex Component Analysis](https://mlsanity.com/paper/2103.10027/) (2021)
- [On Statistical Learning of Simplices: Unmixing Problem Revisited](https://mlsanity.com/paper/1810.07845/) (2018)
- [Optimal estimation of sparse topic models](https://mlsanity.com/paper/2001.07861/) (2020)
- [Sample Complexity Bounds for Learning High-dimensional Simplices in Noisy Regimes](https://mlsanity.com/paper/2209.05953/) (2022)
- [Fundamental Limits of Learning High-dimensional Simplices in Noisy Regimes](https://mlsanity.com/paper/2506.10101/) (2025)

## Abstract (arXiv)

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.