18,202 research outputs found
Maximizing entropy of image models for 2-D constrained coding
This paper considers estimating and maximizing the entropy of two-dimensional (2-D) fields with application to 2-D constrained coding. We consider Markov random fields (MRF), which have a non-causal description, and the special case of Pickard random fields (PRF). The PRF are 2-D causal finite context models, which define stationary probability distributions on finite rectangles and thus allow for calculation of the entropy. We consider two binary constraints and revisit the hard square constraint given by forbidding neighboring 1s and provide novel results for the constraint that no uniform 2 £ 2 squares contains all 0s or all 1s. The maximum values of the entropy for the constraints are estimated and binary PRF satisfying the constraint are characterized and optimized w.r.t. the entropy. The maximum binary PRF entropy is 0.839 bits/symbol for the no uniform squares constraint. The entropy of the Markov random field defined by the 2-D constraint is estimated to be (upper bounded by) 0.8570 bits/symbol using the iterative technique of Belief Propagation on 2 £ 2 finite lattices. Based on combinatorial bounding techniques the maximum entropy for the constraint was determined to be 0.848
A Probabilistic Proof of the Rogers Ramanujan Identities
The asymptotic probability theory of conjugacy classes of the finite general
linear and unitary groups leads to a probability measure on the set of all
partitions of natural numbers. A simple method of understanding these measures
in terms of Markov chains is given and compared with work on the uniform
measure. Elementary probabilistic proofs of the Rogers-Ramanujan identities
follow. As a corollary, the main case of Bailey's lemma is interpreted as
finding eigenvectors of the transition matrix of the Markov chain. It is shown
that the viewpoint of Markov chains extends to quivers.Comment: Final version, to appear in Bull LMS. The one math change is to fix a
typo in the limit in Corollary 2. We also make two historical correction
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