Application of Probabilistic Programming to the Multidimensional Data Compression Problem

从2002年到2003年，研究人员发表的论文内容可分为两部分，分别为：（1）.与Hubert第13问题有关的一个公开问题的解决。Hubert提出的第13问题是，是否任何多个变量的连续函数都可以表示为由多个少变量的连续函数构成的叠加，Kolmogorov和Arnold在大约50年后解决了这个问题。实际上，整函数论问题，询问是否任何多变量整函数可以表示为由多个少变量整函数构成的叠加本研究课题的代表已经成功地解决了这个基于熵理论方法的开放性问题。(2).A本文讨论了叠加表示问题与诺模图理论之间的关系，首先指出了叠加不可表示性的概念是可以分类的。把多元函数的强叠加不可表示性和弱叠加不可表示性分为强叠加不可表示性和弱叠加不可表示性两个概念，而且，如果多元函数是强不可表示的，则不存在任何能估计多元函数值的诺模图，并且，对于某个正整数k，不存在任何能够通过至少k次运算来估计多个变量的函数的值的诺模图，如果它们是弱不可表示的。

The contents of the researchers published during two years from 2002 till 2003 can be classified into two parts, which are satated as follows :(1).The solution to an open problem related to Hubert's 13th problem.The 13th problem formulated by Hubert, which asks if any continuous functions of several variables can be represented as superpositions constructed from several continuous functions of fewer variables, was solved by Kolmogorov and Arnold after about fifty years.Actually, theentire function theoretic problem asking if any entire functions of several variables can be represented as superpositions constructed from several entire functions of fewer variables has remained to be solved.The representative of this research project has succeeded in giving the solution to this open problem which is based on entropy theoretic methods.(2).A relation between the superposition representation proble'm and the theory of nomographs.Primarily, it has been shown that the concept of superposition irrepresentability can be classified. into two concepts which is called strong superposition irrepresentability and weak superposition irepresentability, respectively.Moreover, there does not exist any nomographs enabling to estimate the values of the functions of several variables if they are strongly irrepresentable, and, for a certain positive integer k, there does not exist any nomographs enabling to estimate the values of the functions of several variables by at least k-time operations if they are weakly irrepresentable.