Universal Lossless Compression With Unknown Alphabets—The Average Case

Shamir, G.I.

doi:10.1109/tit.2006.883609

Cited by 23 publications

(58 citation statements)

References 37 publications

(174 reference statements)

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Mentioning

Contrasting

Order By: Relevance

“…Gil Shamir [18] suggests that a bound of similar order can be obtained by properly updating (B12) in [11]. The proof provided in this paper was elaborated independently; both of them use the channel capacity inequality described in Section 3..…”

Section: Theorem 1 For All Integers N Large Enough the Maximin Pattementioning

confidence: 99%

“…It was first introduced byÅberg in [8] as a solution to the multi-alphabet coding problem, where the message x contains only a small subset of the known alphabet A. It was further studied and motivated in a series of articles by Shamir [9][10][11][12] and by Jevtić, Orlitsky, Santhanam and Zhang [13][14][15][16] for practical applications: the alphabet is unknown and has to be transmitted separately anyway (for instance, transmission of a text in an unknown language), or the alphabet is very large in comparison to the message (consider the case of images with k = 2 24 colors, or texts when taking words as the alphabet units).…”

Section: Dictionary and Patternmentioning

confidence: 99%

“…However, Orlistky & et al in [15] and Shamir in [11] proved that for some constants c 1 and c 2 it holds that c 1 n…”

Section: Theoremmentioning

confidence: 99%

“…An important difference appears in the treatment of the quantization, see Equation 2. [11] provides fine relations between the minimax average redundancy and the alphabet size. The approach presented here does not discriminate between alphabet sizes; in a short and elegant proof, it leads to a slightly better bound for infinite alphabets.…”

Section: Theorem 1 For All Integers N Large Enough the Maximin Pattementioning

confidence: 99%

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A Lower-Bound for the Maximin Redundancy in Pattern Coding

Garivier

2009

Entropy

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We show that the maximin average redundancy in pattern coding is eventually larger than 1.84for messages of length n. This improves recent results on pattern redundancy, although it does not fill the gap between known lower-and upper-bounds. The pattern of a string is obtained by replacing each symbol by the index of its first occurrence. The problem of pattern coding is of interest because strongly universal codes have been proved to exist for patterns while universal message coding is impossible for memoryless sources on an infinite alphabet. The proof uses fine combinatorial results on partitions with small summands.

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Section: Theorem 1 For All Integers N Large Enough the Maximin Pattementioning

confidence: 99%

Section: Dictionary and Patternmentioning

confidence: 99%

“…However, Orlistky & et al in [15] and Shamir in [11] proved that for some constants c 1 and c 2 it holds that c 1 n…”

Section: Theoremmentioning

confidence: 99%

Section: Theorem 1 For All Integers N Large Enough the Maximin Pattementioning

confidence: 99%

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A Lower-Bound for the Maximin Redundancy in Pattern Coding

Garivier

2009

Entropy

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“…Initial work on patterns [5], [7], [11], [15], [16], focused on showing diminishing universal compression redundancy rates. The first results on pattern entropy in [11], [14], [16], however, showed that for sufficiently large alphabets, the pattern block entropy must decrease from the i.i.d.…”

mentioning

confidence: 99%

Pattern entropy - revisited

Shamir

2008

2008 Information Theory and Applications Workshop

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Abstract-A pattern of a sequence is a sequence of integer indices with each index describing the order of first occurrence of the respective symbol in the original sequence. Several recent works studied entropy and entropy rate of patterns. Specifically, in a recent paper, tight general bounds on the block entropy of patterns of sequences generated by independent and identically distributed (i.i.d.) sources were derived. In this paper, precise approximations are given to the pattern block entropies for patterns of sequences generated by i.i.d. uniform and monotonic distributions, including distributions over the integers, and the geometric distribution. Numerical non-asymptotic bounds on the pattern block entropies of these distributions are provided even for very short blocks, and even for distributions that have infinite i.i.d. entropy rates. Conditional index entropy is also studied for distributions over smaller alphabets.

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