RUI: Applications of Approximation Theory to Neural Networks and Wavelets

Mhaskar将继续研究神经网络理论中的复杂性问题，以及基于正交多项式系数和目标函数样本的多尺度构造。这些经典方法的变化从近似理论，如可和性方法和多项式不等式将发展，以提供这些明显不同的领域的统一理论。这项工作将应用于正交多项式的数值构造，使用分散数据在球体上的近似，以及系统识别理论。虽然近似理论是一个经典的数学领域，活跃了一百多年，但神经网络和小波领域的技术进步为近似理论的应用提供了许多新的可能性。神经网络和小波在高速并行计算中都很有用，并且自然地涉及到函数的近似。与神经网络一起使用的近似理论技术在阵列天线技术，模式识别和金融时间序列预测的某些应用中产生了显着更好的结果。

Mhaskar will continue his investigations on the complexity problem inthe theory of neural networks and construction of multiscales based onorthogonal polynomial coefficients as well as samples of the targetfunctions. Variations of such classical methods from approximationtheory as summability methods and polynomial inequalities will bedeveloped to provide a unified theory of these apparently diverse areas.The work will be applied to the numerical construction of orthogonalpolynomials, approximation on the sphere using scattered data, and thetheory of system identification.Although approximation theory is a classical area of mathematics, activefor more than a hundred years, technological progress in the areas ofneural networks and wavelets has given rise to many new possibilitiesfor applications of approximation theory. Neural networks and waveletsare both useful in high speed parallel computing, and naturally involvethe approximation of functions. Approximation theory techniques usedwith neural networks have produced dramatically better results incertain applications to array antenna technology, pattern recognition,and financial time series prediction.