8,706 research outputs found
Results on the Redundancy of Universal Compression for Finite-Length Sequences
In this paper, we investigate the redundancy of universal coding schemes on
smooth parametric sources in the finite-length regime. We derive an upper bound
on the probability of the event that a sequence of length , chosen using
Jeffreys' prior from the family of parametric sources with unknown
parameters, is compressed with a redundancy smaller than
for any . Our results also confirm
that for large enough and , the average minimax redundancy provides a
good estimate for the redundancy of most sources. Our result may be used to
evaluate the performance of universal source coding schemes on finite-length
sequences. Additionally, we precisely characterize the minimax redundancy for
two--stage codes. We demonstrate that the two--stage assumption incurs a
negligible redundancy especially when the number of source parameters is large.
Finally, we show that the redundancy is significant in the compression of small
sequences.Comment: accepted in the 2011 IEEE International Symposium on Information
Theory (ISIT 2011
Universal Lossless Compression with Unknown Alphabets - The Average Case
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 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 sub-optimal
low-complexity sequential algorithms for compression of patterns are presented
and their description lengths analyzed, also pointing out that the pattern
average universal description length can decrease below the underlying i.i.d.\
entropy for large enough alphabets.Comment: Revised for IEEE Transactions on Information Theor
A vector quantization approach to universal noiseless coding and quantization
A two-stage code is a block code in which each block of data is coded in two stages: the first stage codes the identity of a block code among a collection of codes, and the second stage codes the data using the identified code. The collection of codes may be noiseless codes, fixed-rate quantizers, or variable-rate quantizers. We take a vector quantization approach to two-stage coding, in which the first stage code can be regarded as a vector quantizer that âquantizesâ the input data of length n to one of a fixed collection of block codes. We apply the generalized Lloyd algorithm to the first-stage quantizer, using induced measures of rate and distortion, to design locally optimal two-stage codes. On a source of medical images, two-stage variable-rate vector quantizers designed in this way outperform standard (one-stage) fixed-rate vector quantizers by over 9 dB. The tail of the operational distortion-rate function of the first-stage quantizer determines the optimal rate of convergence of the redundancy of a universal sequence of two-stage codes. We show that there exist two-stage universal noiseless codes, fixed-rate quantizers, and variable-rate quantizers whose per-letter rate and distortion redundancies converge to zero as (k/2)n -1 log n, when the universe of sources has finite dimension k. This extends the achievability part of Rissanen's theorem from universal noiseless codes to universal quantizers. Further, we show that the redundancies converge as O(n-1) when the universe of sources is countable, and as O(n-1+ϵ) when the universe of sources is infinite-dimensional, under appropriate conditions
Orthogonal Codes for Robust Low-Cost Communication
Orthogonal coding schemes, known to asymptotically achieve the capacity per
unit cost (CPUC) for single-user ergodic memoryless channels with a zero-cost
input symbol, are investigated for single-user compound memoryless channels,
which exhibit uncertainties in their input-output statistical relationships. A
minimax formulation is adopted to attain robustness. First, a class of
achievable rates per unit cost (ARPUC) is derived, and its utility is
demonstrated through several representative case studies. Second, when the
uncertainty set of channel transition statistics satisfies a convexity
property, optimization is performed over the class of ARPUC through utilizing
results of minimax robustness. The resulting CPUC lower bound indicates the
ultimate performance of the orthogonal coding scheme, and coincides with the
CPUC under certain restrictive conditions. Finally, still under the convexity
property, it is shown that the CPUC can generally be achieved, through
utilizing a so-called mixed strategy in which an orthogonal code contains an
appropriate composition of different nonzero-cost input symbols.Comment: 2nd revision, accepted for publicatio
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
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
- âŠ