可扩展最小体积单纯形估计及其非渐近分析

中文摘要 AI 辅助

我们研究从$K$维单纯形内部均匀采样的$N$个独立同分布点来估计该单纯形的问题；观测值是$K+1$个未知原型的凸组合。现有的多项式时间估计器需要立方级的每样本工作量或$O(NK)$的存储，在$N\sim 10^6$--$10^8$时不可行。我们提出DeepMVSA，将最小体积原理以神经隐式形式重新表达：一个轻量级坐标网络生成混合权重，一个三角LU型参数化生成对偶单纯形矩阵，将可训练状态内存降至$O(K^2)$（与$N$无关），每次数据遍历的成本降至$O(NK^2)$。我们证明了局部替代估计器的非渐近样本复杂度界达到多项式时间基准阶；对神经目标的每个全局最小化器给出一个oracle不等式，包含体积膨胀控制和显式收缩偏差；一个条件端到端误差预算，在显式包络事件上分离统计、逼近、优化和包络残差项；以及两点下界：在任何独立于$N$固定的噪声水平$\sigma>0$下，$N^{-1/2}$的缩放在其$N$指数上不可改进。使用多达$N=10^8$合成观测的实验与预测的精度和缩放一致，并在约$10^7$像素的真实场景上展示了可行性。

英文摘要

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.

发表机构

• China University of Geosciences（中国地质大学）

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