Probabilistic Analysis of Large Geometric Structures

本提案研究大型数据集的统计。这些数据集通常出现在应用概率论和基础概率论中，是（拓扑）数据分析、网络理论以及表面估计问题的起点。如何使用数据集来推断确定性曲面的属性？如何使用数据集来得出关于生成数据的对象的结论？产生数据的对象的大小、维度和熵是什么？数据是否有“极端”点和异常值，如果有，有多少？模拟现实世界现象的随机网络的几何特性是什么？网络是否连通，是否“高效”？到目前为止，对大型数据集的研究在很大程度上假设了底层点集的空间独立性，在这种情况下，问题既重要又具有挑战性。我们建议研究这样的模型，以及更现实的模型，其中点不被假定为独立的。这将包括在物理学、计算机科学和运筹学中经常遇到的结构。在随机几何和应用几何概率中出现的问题通常被理解为大型随机几何结构的统计行为，其中“大”意味着随机性涉及越来越多的随机变量。涉及这些结构的问题包括理解具有短距离相互作用的空间依赖项的行为，但复杂的长距离依赖。随机几何结构出现在不同的环境中，包括以下基本例子：(i)相依点的点过程，包括具有行列式、吉布氏或马尔可夫随机场结构的点过程，高斯解析函数的零点和随机Burgers方程的解的零点过程，（ii）拓扑数据分析中的简单复合体，（iii）随机顶点集上的几何网络和几何图，包括在数据融合网络和用于识别数据云的固有维数和熵的最近邻图中产生的几何网络和几何图，（iv）一致估计目标表面的随机曲面；(v)随机数据生成的随机多面体。由大量随机变量生成的多面体的性质在凸几何、算法的平均复杂性、优化和极端统计以及（vi）空间出生生长模型、随机顺序吸附模型中都很有趣。本文对上述大型结构进行统计研究。有了大量的输入，人们可以合理地得出关于大量有趣的统计数据的典型或平均行为的结论。这包括发现均值和方差渐近性以及中心极限定理。

This proposal studies statistics of large data sets. Such data sets typically arise in applied and foundational probability theory and are the starting point for questions in (topological) data analysis, network theory, as well as surface estimation. How does one use a data set to infer properties about a deterministic surface? How does one use a data set to draw conclusions about the objects generating the data? What is the size, dimension, and entropy of the object producing the data? Does the data have `extreme' points and outliers, and if so, how many? What are the geometric properties of random networks which model real world phenomena? Are the networks connected and are they `efficient'? Up to now, the study of large data sets has largely assumed spatial independence of the underlying point sets, where already the questions in this setting are as challenging as they are important. We propose to study such models, but also the more realistic models where points are not assumed independent. This would encompass structures frequently encountered in physics, computer science, and operations research. Questions arising in stochastic geometry and applied geometric probability are often understood in terms of the behavior of statistics of large random geometric structures, where `large' means that the randomness involves a growing number of random variables. Problems involving these structures involve understanding the behavior of spatially dependent terms having short range interactions, but complicated long range dependence. Random geometric structures arise in diverse settings and include these fundamental examples: (i) Point processes of dependent points, including those with determinantal, Gibbsian, or Markov random field structure, zeros of Gaussian analytic functions and zeros of solutions of the stochastic Burgers' equation, (ii) Simplicial complexes in topological data analysis, (iii) Geometric networks and geometric graphs on random vertex sets, including those arising in data fusion networks and nearest neighbor graphs used in discerning intrinsic dimension and entropy of data clouds, (iv) Random surfaces which consistently estimate a target surface, (v) Random polytopes generated by random data. Properties of polytopes generated by a large collection of random variables are of interest in convex geometry, average complexity of algorithms, optimization, and extreme statistics, and(vi) Spatial birth growth models, random sequential adsorption models. This proposal studies statistics of the above-mentioned large structures. With large input, one may reasonably draw conclusions about the typical or average behavior of a large number of interesting statistics. This includes finding mean and variance asymptotics as well as central limit theorems.