Characterizing the Asymptotic Per-Symbol Redundancy of Memoryless Sources over Countable Alphabets in Terms of Single-Letter Marginals

Entropy

Maryam Hosseini¹ 

[1] Department of Electrical Engineering, University of Hawaii at Manoa, Honolulu, HI 96822, USA; E-Mail

关键词: universal compression;    redundancy;    large alphabets;    tightness;    redundancy-capacity theorem;   

DOI  :  10.3390/e16074168

来源: mdpi

【 摘 要 】

The minimum expected number of bits needed to describe a random variable is its entropy, assuming knowledge of the distribution of the random variable. On the other hand, universal compression describes data supposing that the underlying distribution is unknown, but that it belongs to a known set 𝒫 of distributions. However, since universal descriptions are not matched exactly to the underlying distribution, the number of bits they use on average is higher, and the excess over the entropy used is the redundancy. In this paper, we study the redundancy incurred by the universal description of strings of positive integers (ℤ₊), the strings being generated independently and identically distributed (i.i.d.) according an unknown distribution over ℤ₊ in a known collection 𝒫. We first show that if describing a single symbol incurs finite redundancy, then 𝒫 is tight, but that the converse does not always hold. If a single symbol can be described with finite worst-case regret (a more stringent formulation than redundancy above), then it is known that describing length n i.i.d. strings only incurs vanishing (to zero) redundancy per symbol as n increases. On the contrary, we show it is possible that the description of a single symbol from an unknown distribution of 𝒫 incurs finite redundancy, yet the description of length n i.i.d. strings incurs a constant (> 0) redundancy per symbol encoded. We then show a sufficient condition on single-letter marginals, such that length n i.i.d. samples will incur vanishing redundancy per symbol encoded.

【 授权许可】

CC BY   
© 2014 by the authors; licensee MDPI, Basel, Switzerland