Greedy Approximations with Expansions
带有扩展的贪婪近似
基本信息
- 批准号:0554832
- 负责人:
- 金额:$ 11.69万
- 依托单位:
- 依托单位国家:美国
- 项目类别:Standard Grant
- 财政年份:2006
- 资助国家:美国
- 起止时间:2006-06-01 至 2009-08-31
- 项目状态:已结题
- 来源:
- 关键词:
项目摘要
AbstractSparse representations of a function are not only a powerful analytictool but they are utilized in many application areas such as image/signal processing and numerical computation. The backbone of finding sparse representations is the concept of m-term approximation of the target function by the elements of a given system of functions (dictionary). Since the elements of the dictionary used in the m-term approximation are allowed to depend on the function being approximated, this type of approximation, known as nonlinear approximation, is very efficient when the approximants can be found. Nonlinear approximation seeks ways to approximate complicated functions by simple functions using methods that depend nonlinearly on the function being approximated. Recently, a particular kind of nonlinear approximation, namely, greedy approximation attracted a lot of attention in both theoretical and applied settings. Greedy type algorithms proved to be very useful in various applications such as image compression, signal processing, design of neural networks, and the numerical solution of nonlinear partial differential equations. The theory of greedy approximation is emerging now: some convergence results have already been established; many problems remain unsolved. The fundamental question is how to construct good methods (algorithms) of approximation. The purpose of the proposed research is to design and study general nonlinear methods of approximation that are practically realizable. The proposed research will develop algorithms that are provably efficient with respect to convergence and rate of convergence.The goal of the proposed research is to carry out fundamental mathematical and algorithmic study to significantly increase our ability to process (compress, de-noise, etc.) large data sets. The main technique that will be used in achieving this goal is based on nonlinear sparse representations. Understanding how to process large data sets is one of the great scientific challenges of this decade. It is key to designing systems to efficiently analyze data and extract essential information. The scientific discipline, which studies the process of replacing large data by smaller and simpler data, is the approximation theory. It has a myriad of existing and potential applications in both the defense and civilian sectors. For instance, managing large data bases such as security data bases obtained through surveillance requires processing of the data sets in order to speed up extraction of significant features or specific information.
函数的稀疏表示不仅是一种强大的分析工具,而且在图像/信号处理和数值计算等许多应用领域都有广泛的应用. 寻找稀疏表示的支柱是目标函数的m项近似的概念,由给定的函数系统(字典)的元素。 由于m-项近似中使用的字典元素可以依赖于被近似的函数,因此这种类型的近似,称为非线性近似,当可以找到近似时是非常有效的。非线性近似寻求通过简单函数来近似复杂函数的方法,使用非线性依赖于被近似的函数的方法。近年来,一种特殊的非线性逼近,即贪婪逼近,在理论和应用上都引起了广泛的关注。贪婪算法在图像压缩、信号处理、神经网络设计、非线性偏微分方程数值解等领域有着广泛的应用。贪婪逼近理论是一个新兴的理论,一些收敛性结果已经建立,但仍有许多问题没有解决。基本问题是如何构造好的近似方法(算法)。所提出的研究的目的是设计和研究一般的非线性近似方法,实际上是可实现的。拟议的研究将开发算法,证明是有效的收敛性和收敛速度。拟议的研究的目标是进行基本的数学和算法研究,以显着提高我们的处理能力(压缩,去噪等)。大型数据集。将用于实现这一目标的主要技术是基于非线性稀疏表示。理解如何处理大型数据集是本世纪最大的科学挑战之一。它是设计系统以有效分析数据和提取基本信息的关键。近似理论是研究用更小、更简单的数据代替大数据的过程的科学学科。它在国防和民用领域都有无数现有和潜在的应用。例如,管理诸如通过监视获得的安全数据库之类的大型数据库需要处理数据集,以便加速重要特征或特定信息的提取。
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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Vladimir Temlyakov其他文献
Weight-Almost Greedy Bases
- DOI:
10.1134/s0081543818080102 - 发表时间:
2019-03-05 - 期刊:
- 影响因子:0.400
- 作者:
S. J. Dilworth;Denka Kutzarova;Vladimir Temlyakov;Ben Wallis - 通讯作者:
Ben Wallis
Greedy Algorithms with Prescribed Coefficients
- DOI:
10.1007/s00041-006-6033-x - 发表时间:
2007-02-08 - 期刊:
- 影响因子:1.200
- 作者:
Vladimir Temlyakov - 通讯作者:
Vladimir Temlyakov
Vladimir Temlyakov的其他文献
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{{ truncateString('Vladimir Temlyakov', 18)}}的其他基金
Constructive Approximation and Harmonic Analysis
构造近似和调和分析
- 批准号:
1613790 - 财政年份:2016
- 资助金额:
$ 11.69万 - 项目类别:
Standard Grant
Greedy Approximation in Banach Spaces and Compressed Sensing
Banach 空间中的贪婪逼近和压缩感知
- 批准号:
1160841 - 财政年份:2012
- 资助金额:
$ 11.69万 - 项目类别:
Standard Grant
Application of Greedy Approximations in Numerical Integration and Learning Theory
贪心近似在数值积分和学习理论中的应用
- 批准号:
0906260 - 财政年份:2009
- 资助金额:
$ 11.69万 - 项目类别:
Standard Grant
Mathematical Sciences: Multivariate Approximation
数学科学:多元逼近
- 批准号:
9622925 - 财政年份:1996
- 资助金额:
$ 11.69万 - 项目类别:
Standard Grant
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