Scalable method estimates high-dimensional shapes from massive data
Scalable Minimum-Volume Simplex Estimation with Non-asymptotic Analysis
Machine Learning
Summary
Estimating the shape of a high-dimensional simplex (a generalization of a triangle or tetrahedron) from many data points is useful but computationally expensive. The authors present DeepMVSA, a neural network-based approach that greatly reduces memory use and speeds up calculations, making it practical to handle tens or hundreds of millions of points. They also provide mathematical guarantees on how accurate and efficient the method is, with experiments confirming these results even on very large datasets. This approach helps analyze big data where observations come from mixtures of unknown basic components.
What this means in practice
- •For machine learning engineers: Train mixture models efficiently on large datasets with millions of points using minimal memory.
- •For computer vision developers: Estimate complex scene components from pixel data in large images to improve scene understanding and reconstruction.
Authors
Jun LI, Yanlong Guo, Zhaozhao Zeng
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 $N\sim 10^6$--$10^8$. 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 $σ>0$ 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.