Greedy Approximation

PI：弗拉基米尔Temlyakov，南卡罗来纳大学DMS-0200187 翻译后摘要：非线性逼近寻求方法来近似复杂的功能，通过简单的功能，使用的方法，依赖于非线性的功能被近似。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。贪婪逼近理论目前正在兴起：一些基本的收敛性结果已经建立，许多基本问题仍然没有解决。本研究的目的是继续研究贪婪近似。我们建议继续研究贪婪近似的基础上，最近取得了实质性进展。我们建议把重点放在研究贪婪算法的效率方面的冗余系统（字典）。 一方面，Redundant提供了更高的效率方面的近似率的承诺，但另一方面，引起了高度非平凡的理论和practicalproblems。我们注意到冗余系统在理论问题和实际应用中的重要性有充分的理由。逼近论是数学的一个分支，它研究用简单对象代替复杂对象的方法。这一思想在解决真实的世界问题的许多应用中已被证明是卓有成效的。这些应用包括信号处理、图像压缩、金融问题和许多其他问题。 作为模型问题之一，考虑图像压缩。以电视屏幕上的图像（图片）为例。为什么我们要近似它？ 在许多情况下，我们无法传输（或存储在计算机存储器中）图像的整个信息，这可能是因为传输一位信息的成本很高或信道容量有限。这正是应用近似理论可以取得丰硕成果的地方。显然，当我们用其近似值替换图像时，我们会失去图像的质量：我们保留的信息越多，我们所拥有的原始图像的近似度就越好。因此，我们有一个相互作用之间的减少信息和近似的质量。在近似理论中，我们试图找到这个问题的最佳（最优）解决方案。拟议的研究的目的是继续调查的方法的近似 这是由这些类型的应用程序驱动的。我们注意到，非线性近似是在当前的近似理论的前沿。非线性近似是目前唯一的希望，因为要处理真实的实际问题，仍然需要增加可用的近似能力。

PI: Vladimir Temlyakov, University of South CarolinaDMS-0200187 Abstract: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 fundamental convergence results have already been established; many fundamental problems remain unsolved. The purpose of the proposed research is to continue investigations of greedy approximation. We propose to continue to study greedy approximation with regard to bases, where substantial progress has been achieved recently. We propose to place emphasis on studying the efficiency of greedy algorithms with regard to redundant systems (dictionaries). Redundancy on the one hand offers much promise for greater efficiency in terms of approximation rate, but on the other hand gives rise to highly nontrivial theoretical and practicalproblems. We note that there is a solid justification of importance of redundant systems in both theoretical questions and practical applications.Approximation theory is a branch of Mathematics that studies methods of replacing complicated objects by simpler objects. This idea has proved to be fruitful in many applications to the real world problems. Among these applications are signal processing, image compression, financeproblems, and many others. As one of the model problems consider image compression.Take for example an image (picture) on a TV screen. Why should we approximate it? In many cases we cannot afford to transmit (or store in a computer memory) the whole information of an image, perhaps because of a high cost for transmission of a bit of information orlimited channel capacity. This is exactly the point where an application of approximation theory can be fruitful. Clearly, when we replace an image by its approximant we lose the quality of picture: the more information we keep the better approximation to the original image we have. As a result we have an interplay between the reduction of information and the quality of approximation. In approximation theory we try to find the best (optimal) solution to this problem. The purpose of the proposed research is to continue the investigations of methods of approximation which are motivated by these types of applications. We note that nonlinear approximation is at the current frontiers of approximation theory. Nonlinear approximation is the only hope at present for the increase in available approximation power still needed to handle problems of real practical interest.