Mathematical Sciences: RUI: Applications of Wavelet Analysis to Neural Networks

研究者根据所使用的激活函数、层数和神经元数研究了前馈神经网络在不同函数类中逼近函数的逼近能力。特别强调的是放在维度无关的边界和局部逼近的网络。特别是，该研究在小波理论和神经网络之间建立了牢固的联系。研究了通用时间序列预测器和模式分类器的构造。该项目将小波理论与神经网络理论相结合。这两种理论在数据压缩、时间序列预测、目标分类、模式识别和信号处理等领域都有广泛的应用。因此，这两个研究课题在最近的过去蓬勃发展，在很大程度上是相互独立的。这两种理论之间确实存在某些相似之处，因为它们都处理数学上相似的近似过程。研究者进一步探究了这些相似之处。特别是，他研究了通用映射网络的固有能力和局限性，并开发了有效的训练范例。要研究的网络的一个显著特征是，同一个网络可以很容易地训练和再训练，以使用理论上保证的最小数量的神经元来执行各种任务。

The investigator studies the approximation capabilities of afeedforward neural network for approximating functions in different function classes in terms of the activation function used, the number of layers, and the number of neurons. Particular emphasis is placed on dimension-independent bounds and localized approximation by networks. In particular, the research establishes strong connections between the theory of wavelets and neural networks. Applications to the construction of universal time series predictors and pattern classifiers also are studied. The project connects the theory of wavelets with that of neural networks. Both of these theories have numerous applications in such areas as data compression, prediction of time series, target classification, pattern recognition, and signal processing. Both topics of study have therefore flourished during the recent past, for the most part, independently of each other. There do exist certain similarities between the two theories stemming from the fact that they both deal with mathematically similar processes of approximation. The investigator explores these similarities further. In particular, he studies the inherent capabilities and limitations of universal mapping networks and develops efficient training paradigms. A remarkable feature of the networks to be studied is the ease with which the same network can be trained and retrained to perform a variety of tasks using a theoretically guaranteed minimal number of neurons.