Analytic information theory and beyond
Seminar Room 1, Newton Institute
Analytic information theory aims at studying problems of information theory using analytic techniques of computer science and combinatorics. Following Hadamard's precept, we tackle these problems by complex analysis methods such as generating functions, Mellin transform, Fourier series, saddle point method, analytic poissonization and depoissonization, and singularity analysis. This approach lies at the crossroad of computer science and information theory. In this talk, we concentrate on one facet of information theory (i.e., source coding better known as data compression), namely the redundancy rate problem. The redundancy rate problem for a class of sources is the determination of how far the actual code length exceeds the optimal (ideal) code length. In a minimax scenario one finds the is the determination of how far the actual code length exceeds the optimal (ideal) code length. In a minimax scenario one finds the maximal redundancy over all sources within a certain class while in the average scenario one computes the average redundancy over all possible sources. The redundancy rate problem is typical of a situation where second-order asymptotics play a crucial role (since the leading term of the optimal code length is known to be the entropy of the source). This problem is an ideal candidate for ``analytic information theory''. We survey our recent results on the redundancy rate problem obtained by analytic methods. In particular, we present our findings for the redundancy of Shannon codes, Huffman codes, minimax redundancy for memoryless sources, Markov sources, and renewal processes. In the second part of the talk, we discuss the limitations of classical Shannon information theory, and argue that there is need for post-Shannon information theory.