AF: EAGER: Fundamental High-Dimensional Algorithms

高维集合和分布在科学和工程中无处不在。随着现代传感和数据收集方法的发展，现在的瓶颈是对这些复杂集合的分析。在低维情况下运行良好的算法往往不能很好地随着维数的增加而扩展。因此，理解基本问题的复杂性是至关重要的，例如优化（集合中的最佳点）或集成（集合的总质量）或学习（以便能够决定哪些点在集合中，哪些不在），这也是这个项目的动机。随机漫步抽样是处理高维数据的一种重要方法。PI将研究测试几何随机漫步“在飞行中”收敛性的方法。这些方法通过提供接近最优的收敛性经验检验，将显著提高基于马尔可夫链的算法的性能。它们将允许算法具有针对特定输入而不是最坏情况的复杂性。作为一个应用，PI将研究高维凸体和对数凹分布的更快舍入算法。这里开发的方法和技术将对概率和几何以及算法和复杂性的研究人员感兴趣。PI提出了两个特定的函数来测试球在凸体中的收敛性和一个特定的舍入算法。两者在实验中似乎都表现良好。严格地分析它们会带来很大的挑战，因为已知的概率论技术通常只能在马尔可夫链接近平稳分布时才能说明一些问题。这里最基本的问题是：在马尔可夫链收敛之前，它的分布是什么结构？

High-dimensional sets and distributions are ubiquitous in science and engineering. With modern sensing and data collection methods, the bottleneck is now the analysis of such complex sets. Algorithms that work well in low-dimension often do not scale well as the dimension increases. Thus, understanding the complexity of basic problems such as optimization (the best point in a set) or integration (the total mass of a set) or learning (so as to be able to decide which points are in the set and which aren't) is critical, and is the motivation for this project.An important method to handle data in high dimension is sampling by random walks. The PI will investigate methods to test the convergence of Geometric Random Walks "on-the-fly." Such methods would significantly improve the performance of Markov chain based algorithms by providing nearly optimal empirical tests of convergence. They would allow algorithms to have complexity that is tuned to the specific input rather than the worst-case input. As an application, the PI will investigate faster rounding algorithms for high-dimensional convex bodies and log-concave distributions. The methods and techniques developed here will be of interest to researchers in probability and geometry as well as those in algorithms and complexity.The PI proposes two specific functions to test the convergence of the ball walk in a convex body and a specific rounding algorithm. Both seem to perform well in experiments. Analyzing them rigorously presents major challenges because known techniques from probability theory are typically only able to say something about a Markov chain when it is close to its stationary distribution. The fundamental question here is: what structure does the distribution of a Markov chain have, *before* it converges?