Maximal pattern complexity, dual system and pattern recognition

• Authors: Yu-Mei Xue;Teturo Kamae
• Venue: Theoretical Computer Science
• Year: 2012

For a family @W of sets in R^2 and a finite subset S of R^2, let p"@W(S) be the number of distinct sets of the form S@?@w for all @w@?@W. The maximum pattern complexity p"@W^*(k) is the maximum of p"@W(S) among S with #S=k. The S attaining the maximum is considered as the most effective sampling to distinguish the sets in @W. We obtain the exact values or at least the order of p"@W^*(k) in k for various classes @W. We also discuss the dual problem in the case that #@W=~, that is, consider the partition of R^2 generated by a finite family T@?@W. The number of elements in the partition is written as p"R"^"2(T) and p"R"^"2^*(k) is the maximum of p"R"^"2(T) among T with #T=k. Here, p"@W^*(k)=p"R"^"2^*(k) does not hold in general. For the general setting that @W is an infinite subset of A^@S, where A is a finite alphabet, @S is an arbitrary infinite set, and p"@W^*(k)=max"#"S"="k#@W|"S, it is known that the entropy h(@W):=limk-~logp@W*(k)/k exists and takes value in {log1,log2,...,log#A}. In this paper, we prove that the entropy h(@S) of the dual system coincides with h(@W).