266,946 research outputs found
Minimum Description Length codes are critical
In the Minimum Description Length (MDL) principle, learning from the data is
equivalent to an optimal coding problem. We show that the codes that achieve
optimal compression in MDL are critical in a very precise sense. First, when
they are taken as generative models of samples, they generate samples with
broad empirical distributions and with a high value of the relevance, defined
as the entropy of the empirical frequencies. These results are derived for
different statistical models (Dirichlet model, independent and pairwise
dependent spin models, and restricted Boltzmann machines). Second, MDL codes
sit precisely at a second order phase transition point where the symmetry
between the sampled outcomes is spontaneously broken. The order parameter
controlling the phase transition is the coding cost of the samples. The phase
transition is a manifestation of the optimality of MDL codes, and it arises
because codes that achieve a higher compression do not exist. These results
suggest a clear interpretation of the widespread occurrence of statistical
criticality as a characterization of samples which are maximally informative on
the underlying generative process.Comment: 23 pages, 5 figures; Corrected the author name, revised Section 2.2
(Large Deviations of the Universal Codes Exhibit Phase Transitions),
corrected Eq. (89
Minimum Description Length Induction, Bayesianism, and Kolmogorov Complexity
The relationship between the Bayesian approach and the minimum description
length approach is established. We sharpen and clarify the general modeling
principles MDL and MML, abstracted as the ideal MDL principle and defined from
Bayes's rule by means of Kolmogorov complexity. The basic condition under which
the ideal principle should be applied is encapsulated as the Fundamental
Inequality, which in broad terms states that the principle is valid when the
data are random, relative to every contemplated hypothesis and also these
hypotheses are random relative to the (universal) prior. Basically, the ideal
principle states that the prior probability associated with the hypothesis
should be given by the algorithmic universal probability, and the sum of the
log universal probability of the model plus the log of the probability of the
data given the model should be minimized. If we restrict the model class to the
finite sets then application of the ideal principle turns into Kolmogorov's
minimal sufficient statistic. In general we show that data compression is
almost always the best strategy, both in hypothesis identification and
prediction.Comment: 35 pages, Latex. Submitted IEEE Trans. Inform. Theor
LDPC Codes Which Can Correct Three Errors Under Iterative Decoding
In this paper, we provide necessary and sufficient conditions for a
column-weight-three LDPC code to correct three errors when decoded using
Gallager A algorithm. We then provide a construction technique which results in
a code satisfying the above conditions. We also provide numerical assessment of
code performance via simulation results.Comment: 5 pages, 3 figures, submitted to IEEE Information Theory Workshop
(ITW), 200
Mathematical Programming Decoding of Binary Linear Codes: Theory and Algorithms
Mathematical programming is a branch of applied mathematics and has recently
been used to derive new decoding approaches, challenging established but often
heuristic algorithms based on iterative message passing. Concepts from
mathematical programming used in the context of decoding include linear,
integer, and nonlinear programming, network flows, notions of duality as well
as matroid and polyhedral theory. This survey article reviews and categorizes
decoding methods based on mathematical programming approaches for binary linear
codes over binary-input memoryless symmetric channels.Comment: 17 pages, submitted to the IEEE Transactions on Information Theory.
Published July 201
Binary Tree Approach to Scaling in Unimodal Maps
Ge, Rusjan, and Zweifel (J. Stat. Phys. 59, 1265 (1990)) introduced a binary
tree which represents all the periodic windows in the chaotic regime of
iterated one-dimensional unimodal maps. We consider the scaling behavior in a
modified tree which takes into account the self-similarity of the window
structure. A non-universal geometric convergence of the associated superstable
parameter values towards a Misiurewicz point is observed for almost all binary
sequences with periodic tails. There are an infinite number of exceptional
sequences, however, which lead to superexponential scaling. The origin of such
sequences is explained.Comment: 25 pages, plain Te
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