43,992 research outputs found
Complexity Through Nonextensivity
The problem of defining and studying complexity of a time series has
interested people for years. In the context of dynamical systems, Grassberger
has suggested that a slow approach of the entropy to its extensive asymptotic
limit is a sign of complexity. We investigate this idea further by information
theoretic and statistical mechanics techniques and show that these arguments
can be made precise, and that they generalize many previous approaches to
complexity, in particular unifying ideas from the physics literature with ideas
from learning and coding theory; there are even connections of this statistical
approach to algorithmic or Kolmogorov complexity. Moreover, a set of simple
axioms similar to those used by Shannon in his development of information
theory allows us to prove that the divergent part of the subextensive component
of the entropy is a unique complexity measure. We classify time series by their
complexities and demonstrate that beyond the `logarithmic' complexity classes
widely anticipated in the literature there are qualitatively more complex,
`power--law' classes which deserve more attention.Comment: summarizes and extends physics/000707
Measuring sets in infinite groups
We are now witnessing a rapid growth of a new part of group theory which has
become known as "statistical group theory". A typical result in this area would
say something like ``a random element (or a tuple of elements) of a group G has
a property P with probability p". The validity of a statement like that does,
of course, heavily depend on how one defines probability on groups, or,
equivalently, how one measures sets in a group (in particular, in a free
group). We hope that new approaches to defining probabilities on groups
outlined in this paper create, among other things, an appropriate framework for
the study of the "average case" complexity of algorithms on groups.Comment: 22 page
A Short Introduction to Model Selection, Kolmogorov Complexity and Minimum Description Length (MDL)
The concept of overfitting in model selection is explained and demonstrated
with an example. After providing some background information on information
theory and Kolmogorov complexity, we provide a short explanation of Minimum
Description Length and error minimization. We conclude with a discussion of the
typical features of overfitting in model selection.Comment: 20 pages, Chapter 1 of The Paradox of Overfitting, Master's thesis,
Rijksuniversiteit Groningen, 200
On the Numerical Study of the Complexity and Fractal Dimension of CMB Anisotropies
We consider the problem of numerical computation of the Kolmogorov complexity
and the fractal dimension of the anisotropy spots of Cosmic Microwave
Background (CMB) radiation. Namely, we describe an algorithm of estimation of
the complexity of spots given by certain pixel configuration on a grid and
represent the results of computations for a series of structures of different
complexity. Thus, we demonstrate the calculability of such an abstract
descriptor as the Kolmogorov complexity for CMB digitized maps. The correlation
of complexity of the anisotropy spots with their fractal dimension is revealed
as well. This technique can be especially important while analyzing the data of
the forthcoming space experiments.Comment: LATEX, 3 figure
The Thermodynamics of Network Coding, and an Algorithmic Refinement of the Principle of Maximum Entropy
The principle of maximum entropy (Maxent) is often used to obtain prior
probability distributions as a method to obtain a Gibbs measure under some
restriction giving the probability that a system will be in a certain state
compared to the rest of the elements in the distribution. Because classical
entropy-based Maxent collapses cases confounding all distinct degrees of
randomness and pseudo-randomness, here we take into consideration the
generative mechanism of the systems considered in the ensemble to separate
objects that may comply with the principle under some restriction and whose
entropy is maximal but may be generated recursively from those that are
actually algorithmically random offering a refinement to classical Maxent. We
take advantage of a causal algorithmic calculus to derive a thermodynamic-like
result based on how difficult it is to reprogram a computer code. Using the
distinction between computable and algorithmic randomness we quantify the cost
in information loss associated with reprogramming. To illustrate this we apply
the algorithmic refinement to Maxent on graphs and introduce a Maximal
Algorithmic Randomness Preferential Attachment (MARPA) Algorithm, a
generalisation over previous approaches. We discuss practical implications of
evaluation of network randomness. Our analysis provides insight in that the
reprogrammability asymmetry appears to originate from a non-monotonic
relationship to algorithmic probability. Our analysis motivates further
analysis of the origin and consequences of the aforementioned asymmetries,
reprogrammability, and computation.Comment: 30 page
From Knowledge, Knowability and the Search for Objective Randomness to a New Vision of Complexity
Herein we consider various concepts of entropy as measures of the complexity
of phenomena and in so doing encounter a fundamental problem in physics that
affects how we understand the nature of reality. In essence the difficulty has
to do with our understanding of randomness, irreversibility and
unpredictability using physical theory, and these in turn undermine our
certainty regarding what we can and what we cannot know about complex phenomena
in general. The sources of complexity examined herein appear to be channels for
the amplification of naturally occurring randomness in the physical world. Our
analysis suggests that when the conditions for the renormalization group apply,
this spontaneous randomness, which is not a reflection of our limited
knowledge, but a genuine property of nature, does not realize the conventional
thermodynamic state, and a new condition, intermediate between the dynamic and
the thermodynamic state, emerges. We argue that with this vision of complexity,
life, which with ordinary statistical mechanics seems to be foreign to physics,
becomes a natural consequence of dynamical processes.Comment: Phylosophica
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