24,848 research outputs found
Algorithmic Statistics
While Kolmogorov complexity is the accepted absolute measure of information
content of an individual finite object, a similarly absolute notion is needed
for the relation between an individual data sample and an individual model
summarizing the information in the data, for example, a finite set (or
probability distribution) where the data sample typically came from. The
statistical theory based on such relations between individual objects can be
called algorithmic statistics, in contrast to classical statistical theory that
deals with relations between probabilistic ensembles. We develop the
algorithmic theory of statistic, sufficient statistic, and minimal sufficient
statistic. This theory is based on two-part codes consisting of the code for
the statistic (the model summarizing the regularity, the meaningful
information, in the data) and the model-to-data code. In contrast to the
situation in probabilistic statistical theory, the algorithmic relation of
(minimal) sufficiency is an absolute relation between the individual model and
the individual data sample. We distinguish implicit and explicit descriptions
of the models. We give characterizations of algorithmic (Kolmogorov) minimal
sufficient statistic for all data samples for both description modes--in the
explicit mode under some constraints. We also strengthen and elaborate earlier
results on the ``Kolmogorov structure function'' and ``absolutely
non-stochastic objects''--those rare objects for which the simplest models that
summarize their relevant information (minimal sufficient statistics) are at
least as complex as the objects themselves. We demonstrate a close relation
between the probabilistic notions and the algorithmic ones.Comment: LaTeX, 22 pages, 1 figure, with correction to the published journal
versio
Dimension Extractors and Optimal Decompression
A *dimension extractor* is an algorithm designed to increase the effective
dimension -- i.e., the amount of computational randomness -- of an infinite
binary sequence, in order to turn a "partially random" sequence into a "more
random" sequence. Extractors are exhibited for various effective dimensions,
including constructive, computable, space-bounded, time-bounded, and
finite-state dimension. Using similar techniques, the Kucera-Gacs theorem is
examined from the perspective of decompression, by showing that every infinite
sequence S is Turing reducible to a Martin-Loef random sequence R such that the
asymptotic number of bits of R needed to compute n bits of S, divided by n, is
precisely the constructive dimension of S, which is shown to be the optimal
ratio of query bits to computed bits achievable with Turing reductions. The
extractors and decompressors that are developed lead directly to new
characterizations of some effective dimensions in terms of optimal
decompression by Turing reductions.Comment: This report was combined with a different conference paper "Every
Sequence is Decompressible from a Random One" (cs.IT/0511074, at
http://dx.doi.org/10.1007/11780342_17), and both titles were changed, with
the conference paper incorporated as section 5 of this new combined paper.
The combined paper was accepted to the journal Theory of Computing Systems,
as part of a special issue of invited papers from the second conference on
Computability in Europe, 200
On Macroscopic Complexity and Perceptual Coding
The theoretical limits of 'lossy' data compression algorithms are considered.
The complexity of an object as seen by a macroscopic observer is the size of
the perceptual code which discards all information that can be lost without
altering the perception of the specified observer. The complexity of this
macroscopically observed state is the simplest description of any microstate
comprising that macrostate. Inference and pattern recognition based on
macrostate rather than microstate complexities will take advantage of the
complexity of the macroscopic observer to ignore irrelevant noise
Active Virtual Network Management Prediction: Complexity as a Framework for Prediction, Optimization, and Assurance
Research into active networking has provided the incentive to re-visit what
has traditionally been classified as distinct properties and characteristics of
information transfer such as protocol versus service; at a more fundamental
level this paper considers the blending of computation and communication by
means of complexity. The specific service examined in this paper is network
self-prediction enabled by Active Virtual Network Management Prediction.
Computation/communication is analyzed via Kolmogorov Complexity. The result is
a mechanism to understand and improve the performance of active networking and
Active Virtual Network Management Prediction in particular. The Active Virtual
Network Management Prediction mechanism allows information, in various states
of algorithmic and static form, to be transported in the service of prediction
for network management. The results are generally applicable to algorithmic
transmission of information. Kolmogorov Complexity is used and experimentally
validated as a theory describing the relationship among algorithmic
compression, complexity, and prediction accuracy within an active network.
Finally, the paper concludes with a complexity-based framework for Information
Assurance that attempts to take a holistic view of vulnerability analysis
Applying MDL to Learning Best Model Granularity
The Minimum Description Length (MDL) principle is solidly based on a provably
ideal method of inference using Kolmogorov complexity. We test how the theory
behaves in practice on a general problem in model selection: that of learning
the best model granularity. The performance of a model depends critically on
the granularity, for example the choice of precision of the parameters. Too
high precision generally involves modeling of accidental noise and too low
precision may lead to confusion of models that should be distinguished. This
precision is often determined ad hoc. In MDL the best model is the one that
most compresses a two-part code of the data set: this embodies ``Occam's
Razor.'' In two quite different experimental settings the theoretical value
determined using MDL coincides with the best value found experimentally. In the
first experiment the task is to recognize isolated handwritten characters in
one subject's handwriting, irrespective of size and orientation. Based on a new
modification of elastic matching, using multiple prototypes per character, the
optimal prediction rate is predicted for the learned parameter (length of
sampling interval) considered most likely by MDL, which is shown to coincide
with the best value found experimentally. In the second experiment the task is
to model a robot arm with two degrees of freedom using a three layer
feed-forward neural network where we need to determine the number of nodes in
the hidden layer giving best modeling performance. The optimal model (the one
that extrapolizes best on unseen examples) is predicted for the number of nodes
in the hidden layer considered most likely by MDL, which again is found to
coincide with the best value found experimentally.Comment: LaTeX, 32 pages, 5 figures. Artificial Intelligence journal, To
appea
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