AF: Small: Fundamental High-Dimensional Algorithms based on Convex Geometry and Spectral Methods

AF：小：基于凸几何和谱方法的基本高维算法

• 批准号: 1217793
• 负责人: Santosh Vempala
• 金额: $ 42万
• 依托单位: Georgia Tech Research Corporation
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2012
• 资助国家: 美国
• 起止时间: 2012-09-01 至 2016-08-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1217793&HistoricalAwards=false
• 关键词: AF; Small; Fundamental; Dimensional; Algorithms

This project seeks to discover new algorithms and develop algorithmic tools to address fundamental open problems in the theory of algorithms under the following focus topics: (1) Rounding. The power of affine transformations in the design of algorithms and in their analysis, including for solving LP's in strongly polynomial time and approximately sandwiching convex bodies. (2) Learning. Algorithms for learning polyhedra, learning subspace juntas and identifying planted cliques in random graphs. (3) Isoperimetric inequalities. Extensions of Cheeger's method to higher eigenvalues and multi-partitions, and the KLS hyperplane conjecture. (4) Lattices and convex geometry. Optimization and sampling problems over lattices, including: (a) the complexity of integer programming (determining whether a convex body intersects a given lattice), (b) the complexity of cutting plane methods, and (c) conditions under which lattice points in a convex body be sampled efficiently.The problems explored are of a basic nature, and originate from many areas, including optimization (both discrete and continuous), sampling, machine learning and data mining. With the increasing availability of high-dimensional data in important application areas, efficient tools to handle such data are a necessity. This award addresses some of the most basic questions arising from this need. The PI, an active member of the Algorithms and Randomness Center (ARC), served as its founding director and continues in-depth collaborations with scientists from various fields to identify problems and ideas that could play a fundamental role in understanding the complexity of computation. The project will contribute to graduate courses with online notes, textbooks and up-to-date survey articles.

该项目旨在发现新算法并开发算法工具，以解决算法理论中的基本开放问题，重点主题如下：（1）舍入。仿射变换在算法设计和分析中的威力，包括在强多项式时间内求解线性规划和近似夹层凸体。 (2)学习。用于学习多面体、学习子空间军团和识别随机图中植入的派系的算法。 (3) 等周不等式。 Cheeger 方法对更高特征值和多分区的扩展，以及 KLS 超平面猜想。 (4) 格子和凸几何。格子上的优化和采样问题，包括：（a）整数规划的复杂性（确定凸体是否与给定格子相交），（b）割平面方法的复杂性，以及（c）格子中的格点的条件凸体有效采样。所探讨的问题具有基本性质，源于许多领域，包括优化（离散和连续）、采样、机器学习和数据挖掘。随着重要应用领域中高维数据的可用性不断增加，需要有效的工具来处理此类数据。该奖项解决了因这一需求而产生的一些最基本的问题。该 PI 是算法和随机中心 (ARC) 的活跃成员，担任其创始主任，并继续与各个领域的科学家进行深入合作，以确定在理解计算复杂性方面可发挥基础作用的问题和想法。该项目将为研究生课程提供在线笔记、教科书和最新的调查文章。