About adaptive coding on countable alphabets

This paper sheds light on universal coding with respect to classes of memoryless sources over a countable alphabet defined by an envelope function with finite and non-decreasing hazard rate. We prove that the auto-censuring AC code introduced by Bontemps (2011) is adaptive with respect to the collection of such classes. The analysis builds on the tight characterization of universal redundancy rate in terms of metric entropy % of small source classes by Opper and Haussler (1997) and on a careful analysis of the performance of the AC-coding algorithm. The latter relies on non-asymptotic bounds for maxima of samples from discrete distributions with finite and non-decreasing hazard rate

About Adaptive Coding on Countable Alphabets: Max-Stable Envelope Classes

In this paper, we study the problem of lossless universal source coding for stationary memoryless sources on countably infinite alphabets. This task is generally not achievable without restricting the class of sources over which universality is desired. Building on our prior work, we propose natural families of sources characterized by a common dominating envelope. We particularly emphasize the notion of adaptivity, which is the ability to perform as well as an oracle knowing the envelope, without actually knowing it. This is closely related to the notion of hierarchical universal source coding, but with the important difference that families of envelope classes are not discretely indexed and not necessarily nested. Our contribution is to extend the classes of envelopes over which adaptive universal source coding is possible, namely by including max-stable (heavy-tailed) envelopes which are excellent models in many applications, such as natural language modeling. We derive a minimax lower bound on the redundancy of any code on such envelope classes, including an oracle that knows the envelope. We then propose a constructive code that does not use knowledge of the envelope. The code is computationally efficient and is structured to use an {E}xpanding {T}hreshold for {A}uto-{C}ensoring, and we therefore dub it the \textsc{ETAC}-code. We prove that the \textsc{ETAC}-code achieves the lower bound on the minimax redundancy within a factor logarithmic in the sequence length, and can be therefore qualified as a near-adaptive code over families of heavy-tailed envelopes. For finite and light-tailed envelopes the penalty is even less, and the same code follows closely previous results that explicitly made the light-tailed assumption. Our technical results are founded on methods from regular variation theory and concentration of measure

Universal Compression of Power-Law Distributions

English words and the outputs of many other natural processes are well-known to follow a Zipf distribution. Yet this thoroughly-established property has never been shown to help compress or predict these important processes. We show that the expected redundancy of Zipf distributions of order $\alpha>1$ is roughly the $1/\alpha$ power of the expected redundancy of unrestricted distributions. Hence for these orders, Zipf distributions can be better compressed and predicted than was previously known. Unlike the expected case, we show that worst-case redundancy is roughly the same for Zipf and for unrestricted distributions. Hence Zipf distributions have significantly different worst-case and expected redundancies, making them the first natural distribution class shown to have such a difference.Comment: 20 page

Minimax Trees in Linear Time with Applications

A minimax tree is similar to a Huffman tree except that, instead of minimizing the weighted average of the leaves\u27 depths, it minimizes the maximum of any leaf\u27s weight plus its depth. Golumbic (1976) introduced minimax trees and gave a Huffman-like, $O (n log n)$-time algorithm for building them. Drmota and Szpankowski (2002) gave another $O (n log n)$-time algorithm, which takes linear time when the weights are already sorted by their fractional parts. In this paper we give the first linear-time algorithm for building minimax trees for unsorted real weights

Efficient Universal Noiseless Source Codes

Although the existence of universal noiseless variable-rate codes for the class of discrete stationary ergodic sources has previously been established, very few practical universal encoding methods are available. Efficient implementable universal source coding techniques are discussed in this paper. Results are presented on source codes for which a small value of the maximum redundancy is achieved with a relatively short block length. A constructive proof of the existence of universal noiseless codes for discrete stationary sources is first presented. The proof is shown to provide a method for obtaining efficient universal noiseless variable-rate codes for various classes of sources. For memoryless sources, upper and lower bounds are obtained for the minimax redundancy as a function of the block length of the code. Several techniques for constructing universal noiseless source codes for memoryless sources are presented and their redundancies are compared with the bounds. Consideration is given to possible applications to data compression for certain nonstationary sources