Sovable Models of Finite Multi-resolution analysis

具有有限多分辨率函数的学习模型，例如神经网络、高斯混合和有限小波，经常用于模式识别、机器人控制和时间序列预测。然而，它们的数学基础还没有建立起来，因为它们不是线性或非正则模型。在本研究中，我们阐明了两方面的数学性质，(1)函数逼近能力，(2)统计估计效率。(1)众所周知，其模型的函数逼近误差取决于函数拓扑结构。在本研究中，我们提出了一种基于假设目标函数随机取自函数概率测度的方法来澄清函数逼近误差。在此假设的基础上，证明了平均函数逼近误差是由函数系数的协方差或函数的稀疏性决定的。这一结果为多分辨率分析是否有用提供了一个准则。(2)为了澄清统计估计误差，我们证明了学习模型的随机复杂度是由模型的最深奇异点决定的，我们开发了一种基于Sato-Bernstein的b函数和Hironaka的奇异点分辨率的学习效率计算算法。未来的问题是发展一种估计图像和声音的函数概率测度的方法，并建立最大似然法的数学基础。

Learning models with finite multi-resolution functions, for example, neural networks, gaussian mixtures, and finite wavelets, are often used in pattern recognition, robotic control, and time sequence prediction. However, their mathematical foundation has not been established because they are not linear or not regular models. In this research, we clarified the two aspects of their mathematical properties, (1) function approximation abilities, and (2) statistical estimation effciencies.(1) It is well known that function approximation errors by their models depend on the functional topologies. In this research, we proposed a method to clarify the function approximation errors based on the assumption that the target functions are randomly taken from the function probability measures. Based on this assumption, we proved that the average function approximation errors are determined by the covariance of the coeficients of the functions, or the sparseness of the functions. This result shows a critrion whether the multi-resolution analysis is useful or not.(2) In order to clarify the statistical estimation errors, we have shown that the stochastic complexity of the learning model is determined by the deepest singularities of the model, and we developed an algorithm to calculate the learning efficiency based on the Sato-Bernstein's b-function and Hironaka's resolution of singularities.The problems for the future are to depelop a method to estimate the functional probability measure of the images and sounds and to establish the mathematical foundation of the maximum likelihood method.