AF: EAGER: Fundamental High-Dimensional Algorithms

高维集合和分布在科学和工程中无处不在。利用现代传感和数据收集方法，现在的瓶颈是分析这种复杂的集合。在低维中工作良好的算法通常不会随着维度的增加而扩展。因此，理解基本问题的复杂性，如优化（集合中的最佳点）或整合（集合的总质量）或学习（以便能够决定哪些点在集合中，哪些不在集合中）是至关重要的，也是这个项目的动机。处理高维数据的一个重要方法是随机游走采样。PI将研究测试几何随机游走“动态”收敛性的方法。“这种方法将通过提供近乎最佳的收敛经验测试来显着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输入的复杂性。作为应用，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感兴趣的研究人员在概率和几何，以及那些在算法和复杂性。PI提出了两个特定的功能来测试球行走的收敛性在一个凸体和一个特定的舍入算法。两者在实验中表现良好。严格分析它们提出了重大挑战，因为概率论的已知技术通常只能在马尔可夫链接近其平稳分布时才能说明一些问题。这里的基本问题是：马尔可夫链的分布在收敛之前有什么结构？

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?