Results 1  10
of
11
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
Adaptive Coding and Prediction of Sources With Large and Infinite Alphabets
"... Abstract—The problem of predicting a sequence x;x;...generated by a discrete source with unknown statistics is considered. Each letter x is predicted using the information on the word x x 111x only. This problem is of great importance for data compression, because of its use to estimate probability ..."
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Abstract—The problem of predicting a sequence x;x;...generated by a discrete source with unknown statistics is considered. Each letter x is predicted using the information on the word x x 111x only. This problem is of great importance for data compression, because of its use to estimate probability distributions for PPM algorithms and other adaptive codes. On the other hand, such prediction is a classical problem which has received much attention. Its history can be traced back to Laplace. We address the problem where the sequence is generated by an independent and identically distributed (i.i.d.) source with some large (or even infinite) alphabet and suggest a class of new methods of prediction. Index Terms—Adaptive coding, Laplace problem of succession, lossless data compression, prediction of random processes, Shannon entropy, source coding. I.
Minimax Pointwise Redundancy for Memoryless Models over Large Alphabets ∗
"... Abstract—We study the minimax pointwise redundancy of universal coding for memoryless models over large alphabets and present two main results: We first complete studies initiated in Orlitsky and Santhanam [15] deriving precise asymptotics of the minimax pointwise redundancy for all ranges of the al ..."
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Cited by 1 (0 self)
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Abstract—We study the minimax pointwise redundancy of universal coding for memoryless models over large alphabets and present two main results: We first complete studies initiated in Orlitsky and Santhanam [15] deriving precise asymptotics of the minimax pointwise redundancy for all ranges of the alphabet size relative to the sequence length. Second, we consider the pointwise minimax redundancy for a family of models in which some symbol probabilities are fixed. The latter problem leads to a binomial sum for functions with superpolynomial growth. Our findings can be used to approximate numerically the minimax pointwise redundancy for various ranges of the sequence length and the alphabet size. These results are obtained by analytic techniques such as treelike generating functions and the saddle point method. I.
Minimax Redundancy for Large Alphabets
"... Abstract—We study the minimax redundancy of universal coding for large alphabets over memoryless sources and present two main results: We first complete studies initiated in Orlitsky and Santhanam [12] deriving precise asymptotics of the minimax redundancy for all ranges of the alphabet sizes. Secon ..."
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Cited by 1 (0 self)
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Abstract—We study the minimax redundancy of universal coding for large alphabets over memoryless sources and present two main results: We first complete studies initiated in Orlitsky and Santhanam [12] deriving precise asymptotics of the minimax redundancy for all ranges of the alphabet sizes. Second, we consider the minimax redundancy of a source model in which some symbol probabilities are fixed. The latter model leads to an interesting binomial sum asymptotics with superexponential growth functions. Our findings could be used to approximate numerically the minimax redundancy for various ranges of the sequence length and the alphabet size. These results are obtained by analytic techniques such as treelike generating functions and the saddle point method. I.
Universal Noiseless Compression for Noisy Data
"... Abstract — We study universal compression for discrete data sequences that were corrupted by noise. We show that while, as expected, there exist many cases in which the entropy of these sequences increases from that of the original data, somewhat surprisingly and counterintuitively, universal codin ..."
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Abstract — We study universal compression for discrete data sequences that were corrupted by noise. We show that while, as expected, there exist many cases in which the entropy of these sequences increases from that of the original data, somewhat surprisingly and counterintuitively, universal coding redundancy of such sequences cannot increase compared to the original data. We derive conditions that guarantee that this redundancy does not decrease asymptotically (in first order) from the original sequence redundancy in the stationary memoryless case. We then provide bounds on the redundancy for coding finite length (large) noisy blocks generated by stationary memoryless sources and corrupted by some specific memoryless channels. Finally, we propose a sequential probability estimation method that can be used to compress binary data corrupted by some noisy channel. While there is much benefit in using this method in compressing short blocks of noise corrupted data, the new method is more general and allows sequential compression of binary sequences for which the probability of a bit is known to be limited within any given interval (not necessarily between 0 and 1). Additionally, this method has many different applications, including, prediction, sequential channel estimation, and others. I.
Algorithms, Combinatorics, Information, and Beyond
, 2012
"... Shannon information theory aims at finding fundamental limits for storage and communication, including ratesof convergenceto these limits. Indeed, many interesting information theoretic phenomena seem to appear in the second order asymptotics. So we first discuss precise analysis of the minimax redu ..."
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Shannon information theory aims at finding fundamental limits for storage and communication, including ratesof convergenceto these limits. Indeed, many interesting information theoretic phenomena seem to appear in the second order asymptotics. So we first discuss precise analysis of the minimax redundancy that can be viewed as a measure of learnable or useful information. Then we highlight Markov types unveiling some interesting connections to combinatorics of graphical enumeration and linear Diophantine equations. Next we turn our attention to structural compression of graphical objects, proposing a compression algorithm achieving the lower bound represented by the structural entropy. These results are obtained using tools of analytic combinatorics and analysis of algorithms, known also as analytic information theory. Finally, we argue that perhaps information theory needs to be broadened if it is to meet today’s challenges beyond its original goals (of traditional communication) in biology, economics, modern communication, and knowledge extraction. One of the essential components of this perspective is to continue building foundations in better understanding of temporal, spatial, structural and semantic information in dynamic networks with limited resources. Recently, the National Science Foundation has established the first Science and Technology Center on Science of Information (CSoI) to address these challenges and develop tools to move beyond our current understanding of information flow in communication and storage systems. 1
Universal Source Coding for Monotonic and Fast Decaying Monotonic Distributions ∗
, 704
"... We study universal compression of sequences generated by monotonic distributions. We show that for a monotonic distribution over an alphabet of size k, each probability parameter costs essentially 0.5 log(n/k 3) bits, where n is the coded sequence length, as long as k = o(n 1/3). Otherwise, for k = ..."
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We study universal compression of sequences generated by monotonic distributions. We show that for a monotonic distribution over an alphabet of size k, each probability parameter costs essentially 0.5 log(n/k 3) bits, where n is the coded sequence length, as long as k = o(n 1/3). Otherwise, for k = O(n), the total average sequence redundancy is O(n1/3+ε) bits overall. We then show that there exists a subclass of monotonic distributions over infinite alphabets for which redundancy of O(n1/3+ε) bits overall is still achievable. This class contains fast decaying distributions, including many distributions over the integers and geometric distributions. For some slower decays, including other distributions over the integers, redundancy of o(n) bits overall is achievable, where a method to compute specific redundancy rates for such distributions is derived. The results are specifically true for finite entropy monotonic distributions. Finally, we study individual sequence redundancy behavior assuming a sequence is governed by a monotonic distribution. We show that for sequences whose empirical distributions are monotonic, individual redundancy bounds similar to those in the average case can be obtained. However, even if the monotonicity in the empirical distribution is violated, diminishing per symbol individual sequence redundancies with respect to the monotonic maximum likelihood description length may still be achievable.
On Information Divergence Measures and a Unified Typicality
"... Strong typicality, which is more powerful for theorem proving than weak typicality, can be applied to finite alphabets only, while weak typicality can be applied to countable alphabets. In this paper, the relation between typicality and information divergence measures is discussed. The new definitio ..."
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Strong typicality, which is more powerful for theorem proving than weak typicality, can be applied to finite alphabets only, while weak typicality can be applied to countable alphabets. In this paper, the relation between typicality and information divergence measures is discussed. The new definition of information divergence measure in this paper leads to the definition of a unified typicality for finite or countably infinite alphabets which is stronger than both weak typicality and strong typicality. Unified typicality retains the asymptotic equipartition property and the structural properties of strong typicality, and it can potentially be used to generalize those theorems which are previously established by strong typicality to countable alphabets. The applications in ratedistortion theory and multisource network coding problems are discussed. I.
President’s Column
"... The writing of this column has been marked by many different emotions. When I began composing my message, I was thinking about continuing our reflection on our Society in the context of our IEEE review and of the upcom ing ISIT. Having attended the TAB meeting in February, where I attended our Trans ..."
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The writing of this column has been marked by many different emotions. When I began composing my message, I was thinking about continuing our reflection on our Society in the context of our IEEE review and of the upcom ing ISIT. Having attended the TAB meeting in February, where I attended our Transactions ’ glowing review, and being in the midst of preparing for ISIT in Cambridge, I was trying to distill for this column the promises and challenges that lie before us. Before I was able to commit my thoughts to text, the untimely death of our colleague