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Precise Minimax Redundancy and Regret
 IEEE TRANS. INFORMATION THEORY
, 2004
"... Recent years have seen a resurgence of interest in redundancy of lossless coding. The redundancy (regret) of universal xed{to{variable length coding for a class of sources determines by how much the actual code length exceeds the optimal (ideal over the class) code length. In a minimax scenario ..."
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Cited by 33 (13 self)
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Recent years have seen a resurgence of interest in redundancy of lossless coding. The redundancy (regret) of universal xed{to{variable length coding for a class of sources determines by how much the actual code length exceeds the optimal (ideal over the class) code length. In a minimax scenario one nds the best code for the worst source either in the worst case (called also maximal minimax) or on average. We rst study the worst case minimax redundancy over a class of stationary ergodic sources and replace Shtarkov's bound by an exact formula. Among others, we prove that a generalized Shannon code minimizes the worst case redundancy, derive asymptotically its redundancy, and establish some general properties. This allows us to obtain precise redundancy rates for memoryless, Markov and renewal sources. For example, we derive the exact constant of the redundancy rate for memoryless and Markov sources by showing that an integer nature of coding contributes log(log m=(m 1))= log m+ o(1) where m is the size of the alphabet. Then we deal with the average minimax redundancy and regret. Our approach
Universal compression of memoryless sources over unknown alphabets
 IEEE TRANSACTIONS ON INFORMATION THEORY
, 2004
"... It has long been known that the compression redundancy of independent and identically distributed (i.i.d.) strings increases to infinity as the alphabet size grows. It is also apparent that any string can be described by separately conveying its symbols, and its pattern—the order in which the symbol ..."
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Cited by 32 (10 self)
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It has long been known that the compression redundancy of independent and identically distributed (i.i.d.) strings increases to infinity as the alphabet size grows. It is also apparent that any string can be described by separately conveying its symbols, and its pattern—the order in which the symbols appear. Concentrating on the latter, we show that the patterns of i.i.d. strings over all, including infinite and even unknown, alphabets, can be compressed with diminishing redundancy, both in block and sequentially, and that the compression can be performed in linear time. To establish these results, we show that the number of patterns is the Bell number, that the number of patterns with a given number of symbols is the Stirling number of the second kind, and that the redundancy of patterns can be bounded using results of Hardy and Ramanujan on the number of integer partitions. The results also imply an asymptotically optimal solution for the GoodTuring probabilityestimation problem.
Limit results on pattern entropy
 IEEE Trans. Inf. Theory
, 2006
"... We determine the entropy rate of patterns of certain random processes, bound the speed at which the persymbol pattern entropy converges to this rate, and show that patterns satisfy an asymptotic equipartition property. To derive some of these results we upper bound the probability that the n ′ th v ..."
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Cited by 15 (3 self)
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We determine the entropy rate of patterns of certain random processes, bound the speed at which the persymbol pattern entropy converges to this rate, and show that patterns satisfy an asymptotic equipartition property. To derive some of these results we upper bound the probability that the n ′ th variable in a random process differs from all preceding ones.
Universal lossless compression with unknown alphabets  The average case
, 2006
"... Universal compression of patterns of sequences generated by independently identically distributed (i.i.d.) sources with unknown, possibly large, alphabets is investigated. A pattern is a sequence of indices that contains all consecutive indices in increasing order of first occurrence. If the alphabe ..."
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Cited by 11 (3 self)
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Universal compression of patterns of sequences generated by independently identically distributed (i.i.d.) sources with unknown, possibly large, alphabets is investigated. A pattern is a sequence of indices that contains all consecutive indices in increasing order of first occurrence. If the alphabet of a source that generated a sequence is unknown, the inevitable cost of coding the unknown alphabet symbols can be exploited to create the pattern of the sequence. This pattern can in turn be compressed by itself. It is shown that if the alphabet size k is essentially small, then the average minimax and maximin redundancies as well as the redundancy of every code for almost every source, when compressing a pattern, consist of at least 0.5 log ( n/k 3) bits per each unknown probability parameter, and if all alphabet letters are likely to occur, there exist codes whose redundancy is at most 0.5 log ( n/k 2) bits per each unknown probability parameter, where n is the length of the data sequences. Otherwise, if the alphabet is large, these redundancies are essentially at least O ( n −2/3) bits per symbol, and there exist codes that achieve redundancy of essentially O ( n −1/2) bits per symbol. Two suboptimal lowcomplexity sequential algorithms for compression of patterns are presented and their description lengths
A lower bound on compression of unknown alphabets
 Theoret. Comput. Sci
, 2005
"... Many applications call for universal compression of strings over large, possibly infinite, alphabets. However, it has long been known that the resulting redundancy is infinite even for i.i.d. distributions. It was recently shown that the redudancy of the strings ’ patterns, which abstract the values ..."
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Cited by 10 (3 self)
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Many applications call for universal compression of strings over large, possibly infinite, alphabets. However, it has long been known that the resulting redundancy is infinite even for i.i.d. distributions. It was recently shown that the redudancy of the strings ’ patterns, which abstract the values of the symbols, retaining only their relative precedence, is sublinear in the blocklength n, hence the persymbol redundancy diminishes to zero. In this paper we show that pattern redundancy is at least (1.5 log 2 e) n 1/3 bits. To do so, we construct a generating function whose coefficients lower bound the redundancy, and use Hayman’s saddlepoint approximation technique to determine the coefficients ’ asymptotic behavior. 1
On the entropy rate of pattern processes
 Proceedings of the 2005 Data Compression Conference, Snowbird
, 2005
"... We study the entropy rate of pattern sequences of stochastic processes, and its relationship to the entropy rate of the original process. We give a complete characterization of this relationship for i.i.d. processes over arbitrary alphabets, stationary ergodic processes over discrete alphabets, and ..."
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Cited by 6 (0 self)
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We study the entropy rate of pattern sequences of stochastic processes, and its relationship to the entropy rate of the original process. We give a complete characterization of this relationship for i.i.d. processes over arbitrary alphabets, stationary ergodic processes over discrete alphabets, and a broad family of stationary ergodic processes over uncountable alphabets. For cases where the entropy rate of the pattern process is infinite, we characterize the possible growth rate of the block entropy. 1
Universal compression of Markov and related sources over arbitrary alphabets
 IEEE TRANSACTIONS ON INFORMATION THEORY
, 2006
"... Recent work has considered encoding a string by separately conveying its symbols and its pattern—the order in which the symbols appear. It was shown that the patterns of i.i.d. strings can be losslessly compressed with diminishing persymbol redundancy. In this paper the pattern redundancy of distri ..."
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Cited by 3 (2 self)
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Recent work has considered encoding a string by separately conveying its symbols and its pattern—the order in which the symbols appear. It was shown that the patterns of i.i.d. strings can be losslessly compressed with diminishing persymbol redundancy. In this paper the pattern redundancy of distributions with memory is considered. Close lower and upper bounds are established on the pattern redundancy of strings generated by Hidden Markov Models with a small number of states, showing in particular that their persymbol pattern redundancy diminishes with increasing string length. The upper bounds are obtained by analyzing the growth rate of the number of multidimensional integer partitions, and the lower bounds, using Hayman’s Theorem.
A Universal Compression Perspective of Smoothing
"... We analyze smoothing algorithms from a universalcompression perspective. Instead of evaluating their performance on an empirical sample, we analyze their performance on the most inconvenient sample possible. Consequently the performance of the algorithm can be guaranteed even on unseen data. We sho ..."
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Cited by 1 (0 self)
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We analyze smoothing algorithms from a universalcompression perspective. Instead of evaluating their performance on an empirical sample, we analyze their performance on the most inconvenient sample possible. Consequently the performance of the algorithm can be guaranteed even on unseen data. We show that universal compression bounds can explain the empirical performance of several smoothing methods. We also describe a new interpolated additive smoothing algorithm, and show that it has lower training complexity and better compression performance than existing smoothing techniques. Key words: Language modeling, universal compression, smoothing 1
Patterns of i.i.d. Sequences and Their Entropy Part II: Bounds for Some Distributions ∗
, 711
"... 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. In a recent paper, tight general bounds on the block entropy of patterns of sequences generated by independent and identically distribu ..."
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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. 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 provided for 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 bounds on the pattern block entropies of these distributions are provided even for very short blocks. Tight bounds are obtained even for distributions that have infinite i.i.d. entropy rates. The approximations are obtained using general bounds and their derivation techniques. Conditional index entropy is also studied for distributions over smaller alphabets. Index Terms: patterns, monotonic distributions, uniform distributions, entropy.