Optimization over the fixed point set of nonexpansive mapping and its application

非扩张映射不动点集的优化及其应用

基本信息

批准号：
10650350
负责人：
YAMADA Isao
金额：
$ 0.51万
依托单位：
Tokyo Institute of Technology
依托单位国家：
日本
项目类别：
Grant-in-Aid for Scientific Research (C)
财政年份：
1998
资助国家：
日本
起止时间：
1998 至 1999
项目状态：
已结题

来源：
https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-10650350/
关键词：
Hybrid steepest descent method fixed point nonexpansive mapping convex projection POCS blind image deconvolution inverse problem neural network

项目摘要

The convex projection algorithm is a class of algorithms finding a point, with convex projections, in the intersection of multiple closed convex sets. The basic idea of the algorithms originated from J. von Neumann's alternating projection in 1933. Although the point obtained by the method is only guaranteed to belong to the intersection of given closed convex sets, the remarkable effect and the universal applicability of the simple algorithms have been commonly recognized in many branches of applied mathematical, physical, computer sciences and engineerings since the algorithm POCS was successfully applied to the image restoration problem by D.C. Youla and H. Webb in 1982.In this research project, we develop a new algorithm mamed Hybrid steepest descent method that minimizes a given convex cost function over the fixed point set of a nonexpansive mapping in a real Hilbert space. The nonexpansive mapping is extremly general class of mappings including the convex projection. By this great generality, many open problems, in signal processing, not handled by the standard convex projection technique have become resolved. Indeed we successfully applied the method to the following important problems:1. Approximation of convexly constrained pseudoinverse operator,2. Constrained least squares design of M-D FIR filter,3. Design of two channel linear phase FIR QMF banks,4. Set-theoretic blind image deconvolution problem,5. Design of associative memory neural network to recall nearest pattern from input.

凸投影算法是一类在多个闭凸集的交集中用凸投影求点的算法。该算法的基本思想源于J. von Neumann 1933年的交替投影。虽然该方法得到的点只能保证属于给定的闭凸集的交集，但其显着的效果和简单算法的普遍适用性已经自 1982 年 D.C. Youla 和 H. Webb 成功地将 POCS 算法应用于图像恢复问题以来，POCS 算法在应用数学、物理、计算机科学和工程的许多分支中得到了普遍认可。开发一种名为混合最速下降法的新算法，该算法可最小化真实希尔伯特空间中非扩张映射的定点集上给定的凸成本函数。非扩张映射是包括凸投影在内的极其通用的映射类型。通过这种巨大的通用性，信号处理中标准凸投影技术无法处理的许多开放问题已经得到解决。事实上，我们成功地将该方法应用于以下重要问题： 1．凸约束伪逆算子的逼近，2． M-D FIR滤波器的约束最小二乘设计，3．两通道线性相位FIR QMF组的设计，4．集合论盲图像反卷积问题，5．设计联想记忆神经网络以从输入中回忆最近的模式。