Combinatorial and Probabilistic Approach to Geometric Functional Analysis and Applications
几何泛函分析和应用的组合和概率方法
基本信息
- 批准号:0401032
- 负责人:
- 金额:$ 9.48万
- 依托单位:
- 依托单位国家:美国
- 项目类别:Continuing Grant
- 财政年份:2004
- 资助国家:美国
- 起止时间:2004-07-01 至 2007-06-30
- 项目状态:已结题
- 来源:
- 关键词:
项目摘要
AbstractThe aim of the project is to develop a new combinatorial andprobabilistic approach to geometric functional analysis and itsapplications. Some of new major ideas are expected to come from theconcept of the combinatorial dimension, which is a general form of theclassical Vapnik-Chervonenkis dimension. Arising from logic, probabilitytheory and computer science, the use of the combinatorial dimensionlooks very promising also in a wide range of areas including geometricfunctional analysis (finding nice sections of convex bodies), convexgeometry (study of polytopes), discrete geometry (counting integerpoints and cells in sets) and extremal combinatorics. This newcombinatorial and probabilistic method is aimed at one of the hardestproblems in the theory of empirical processes - describe the classes offunctions for which the Central Limit Theorem holds uniformly. Newaspects of the celebrated concentration of measure phenomenon will alsobe studied by a combination of probabilistic and purely geometric ideas.This might give an insight into relationships between random anddeterministic structures in geometric functional analysis, as well as anew view of local versus global asymptotic convex geometries.Probabilistic approach will also be developed for problems of findingnice submatrices of large matrices, which arise in functional andharmonic analysis as well as in computer science.The project opens new connections between functional analysis,combinatorics, probability, convex geometry and applied mathematics. Thecelebrated "probabilistic method" along with deterministiccombinatorial, geometric and analytic methods will merge into onemachinery, which may expand our knowledge on therelationships between chaos and pattern that arise in a variety ofhigh-dimensional structures in pure mathematics and in computer science. From the practical point of view, the results expected from this machineryinclude justification of algorithms in machine learning, development ofalgorithms for storage of large amounts of data and for datatransmission (such as error correction codes).
本项目的目的是发展一种新的组合概率方法来研究几何泛函分析及其应用。组合维数是经典Vapnik-Chervonenkis维数的一般形式,它的提出将带来一些新的重要思想。产生于逻辑、概率论和计算机科学,组合维数的使用在广泛的领域也很有前途,包括几何泛函分析(找到凸体的好部分)、凸几何(研究多面体)、离散几何(计算集合中的整数点和单元格)和极值组合学。这种新的组合和概率的方法是针对一个最难的问题,在理论的经验过程-描述类的功能,其中心极限定理保持一致。著名的测度集中现象的新方面也将通过概率和纯几何思想的结合来研究。这可能会使人们深入了解几何泛函分析中随机和确定性结构之间的关系,以及局部与全局渐近凸几何的新观点。概率方法也将发展为寻找大矩阵的好子矩阵的问题,该项目在泛函分析、组合数学、概率论、凸几何和应用数学之间建立了新的联系。这种被称为“概率方法”的方法沿着与确定性的组合方法、几何方法和分析方法将融合成一种方法,这可能会扩展我们对纯数学和计算机科学中各种高维结构中出现的混沌和模式之间关系的认识。从实践的角度来看,这台机器的预期结果包括机器学习算法的合理性,用于存储大量数据和数据传输(如纠错码)的算法的开发。
项目成果
期刊论文数量(0)
专著数量(0)
科研奖励数量(0)
会议论文数量(0)
专利数量(0)
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Roman Vershynin其他文献
Are most Boolean functions determined by low frequencies?
大多数布尔函数是由低频决定的吗?
- DOI:
- 发表时间:
2024 - 期刊:
- 影响因子:0
- 作者:
Roman Vershynin - 通讯作者:
Roman Vershynin
Hamiltonicity of Sparse Pseudorandom Graphs
稀疏伪随机图的哈密顿性
- DOI:
- 发表时间:
2024 - 期刊:
- 影响因子:0
- 作者:
Asaf Ferber;Jie Han;Dingjia Mao;Roman Vershynin - 通讯作者:
Roman Vershynin
The quarks of attention: Structure and capacity of neural attention building blocks
注意力的夸克:神经注意力构建模块的结构与容量
- DOI:
10.1016/j.artint.2023.103901 - 发表时间:
2023-06-01 - 期刊:
- 影响因子:4.600
- 作者:
Pierre Baldi;Roman Vershynin - 通讯作者:
Roman Vershynin
LECTURES ON FUNCTIONAL ANALYSIS
- DOI:
- 发表时间:
2010 - 期刊:
- 影响因子:0
- 作者:
Roman Vershynin - 通讯作者:
Roman Vershynin
Metric geometry of the privacy-utility tradeoff
隐私与效用权衡的度量几何
- DOI:
- 发表时间:
2024 - 期刊:
- 影响因子:0
- 作者:
M. Boedihardjo;T. Strohmer;Roman Vershynin - 通讯作者:
Roman Vershynin
Roman Vershynin的其他文献
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{{ truncateString('Roman Vershynin', 18)}}的其他基金
High-Dimensional Probability for High-Dimensional Data
高维数据的高维概率
- 批准号:
1954233 - 财政年份:2020
- 资助金额:
$ 9.48万 - 项目类别:
Continuing Grant
Collaborative Research: A Mathematical Framework for Generating Synthetic Data
协作研究:生成综合数据的数学框架
- 批准号:
2027299 - 财政年份:2020
- 资助金额:
$ 9.48万 - 项目类别:
Standard Grant
Geometric functional analysis, random matrices and applications
几何泛函分析、随机矩阵及其应用
- 批准号:
1265782 - 财政年份:2013
- 资助金额:
$ 9.48万 - 项目类别:
Continuing Grant
Non-asymptotic problems on random operators in geometric functional analysis and applications
几何泛函分析中随机算子的非渐近问题及其应用
- 批准号:
1001829 - 财政年份:2010
- 资助金额:
$ 9.48万 - 项目类别:
Continuing Grant
FRG: Collaborative Research: Fourier analytic and probabilistic methods in geometric functional analysis and convexity
FRG:协作研究:几何泛函分析和凸性中的傅里叶分析和概率方法
- 批准号:
0918623 - 财政年份:2008
- 资助金额:
$ 9.48万 - 项目类别:
Standard Grant
FRG: Collaborative Research: Fourier analytic and probabilistic methods in geometric functional analysis and convexity
FRG:协作研究:几何泛函分析和凸性中的傅里叶分析和概率方法
- 批准号:
0652617 - 财政年份:2007
- 资助金额:
$ 9.48万 - 项目类别:
Standard Grant
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