159 research outputs found
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
Universal Coding on Infinite Alphabets: Exponentially Decreasing Envelopes
This paper deals with the problem of universal lossless coding on a countable
infinite alphabet. It focuses on some classes of sources defined by an envelope
condition on the marginal distribution, namely exponentially decreasing
envelope classes with exponent . The minimax redundancy of
exponentially decreasing envelope classes is proved to be equivalent to
. Then a coding strategy is proposed, with
a Bayes redundancy equivalent to the maximin redundancy. At last, an adaptive
algorithm is provided, whose redundancy is equivalent to the minimax redundanc
About Adaptive Coding on Countable Alphabets: Max-Stable Envelope Classes
In this paper, we study the problem of lossless universal source coding for
stationary memoryless sources on countably infinite alphabets. This task is
generally not achievable without restricting the class of sources over which
universality is desired. Building on our prior work, we propose natural
families of sources characterized by a common dominating envelope. We
particularly emphasize the notion of adaptivity, which is the ability to
perform as well as an oracle knowing the envelope, without actually knowing it.
This is closely related to the notion of hierarchical universal source coding,
but with the important difference that families of envelope classes are not
discretely indexed and not necessarily nested.
Our contribution is to extend the classes of envelopes over which adaptive
universal source coding is possible, namely by including max-stable
(heavy-tailed) envelopes which are excellent models in many applications, such
as natural language modeling. We derive a minimax lower bound on the redundancy
of any code on such envelope classes, including an oracle that knows the
envelope. We then propose a constructive code that does not use knowledge of
the envelope. The code is computationally efficient and is structured to use an
{E}xpanding {T}hreshold for {A}uto-{C}ensoring, and we therefore dub it the
\textsc{ETAC}-code. We prove that the \textsc{ETAC}-code achieves the lower
bound on the minimax redundancy within a factor logarithmic in the sequence
length, and can be therefore qualified as a near-adaptive code over families of
heavy-tailed envelopes. For finite and light-tailed envelopes the penalty is
even less, and the same code follows closely previous results that explicitly
made the light-tailed assumption. Our technical results are founded on methods
from regular variation theory and concentration of measure
Coding on countably infinite alphabets
33 pagesInternational audienceThis paper describes universal lossless coding strategies for compressing sources on countably infinite alphabets. Classes of memoryless sources defined by an envelope condition on the marginal distribution provide benchmarks for coding techniques originating from the theory of universal coding over finite alphabets. We prove general upper-bounds on minimax regret and lower-bounds on minimax redundancy for such source classes. The general upper bounds emphasize the role of the Normalized Maximum Likelihood codes with respect to minimax regret in the infinite alphabet context. Lower bounds are derived by tailoring sharp bounds on the redundancy of Krichevsky-Trofimov coders for sources over finite alphabets. Up to logarithmic (resp. constant) factors the bounds are matching for source classes defined by algebraically declining (resp. exponentially vanishing) envelopes. Effective and (almost) adaptive coding techniques are described for the collection of source classes defined by algebraically vanishing envelopes. Those results extend ourknowledge concerning universal coding to contexts where the key tools from parametric inferenc
Coding on countably infinite alphabets
This paper describes universal lossless coding strategies for compressing
sources on countably infinite alphabets. Classes of memoryless sources defined
by an envelope condition on the marginal distribution provide benchmarks for
coding techniques originating from the theory of universal coding over finite
alphabets. We prove general upper-bounds on minimax regret and lower-bounds on
minimax redundancy for such source classes. The general upper bounds emphasize
the role of the Normalized Maximum Likelihood codes with respect to minimax
regret in the infinite alphabet context. Lower bounds are derived by tailoring
sharp bounds on the redundancy of Krichevsky-Trofimov coders for sources over
finite alphabets. Up to logarithmic (resp. constant) factors the bounds are
matching for source classes defined by algebraically declining (resp.
exponentially vanishing) envelopes. Effective and (almost) adaptive coding
techniques are described for the collection of source classes defined by
algebraically vanishing envelopes. Those results extend ourknowledge concerning
universal coding to contexts where the key tools from parametric inferenceComment: 33 page
Universal Compression of Power-Law Distributions
English words and the outputs of many other natural processes are well-known
to follow a Zipf distribution. Yet this thoroughly-established property has
never been shown to help compress or predict these important processes. We show
that the expected redundancy of Zipf distributions of order is
roughly the power of the expected redundancy of unrestricted
distributions. Hence for these orders, Zipf distributions can be better
compressed and predicted than was previously known. Unlike the expected case,
we show that worst-case redundancy is roughly the same for Zipf and for
unrestricted distributions. Hence Zipf distributions have significantly
different worst-case and expected redundancies, making them the first natural
distribution class shown to have such a difference.Comment: 20 page
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