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On the Exact Constant in the Quantitative Steinitz Theorem in the Plane*
"... Abstract. We determine the maximal value of r with the following property. If the convex hull of a set in R 2 contains a unit circle B, then a subset of at most four points can be selected so that the convex hull of this subset contains the circle of radius r concentric with B. That the result is sh ..."
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Abstract. We determine the maximal value of r with the following property. If the convex hull of a set in R 2 contains a unit circle B, then a subset of at most four points can be selected so that the convex hull of this subset contains the circle of radius r concentric with B. That the result is sharp is shown by the example when the original set is the set of vertices of a regular pentagon circumscribed around B. 1.
A QUANTITATIVE STEINITZ THEOREM FOR PLANE TRIANGULATIONS
, 2013
"... We give a new proof of Steinitz's classical theorem in the case of plane triangulations, which allows us to obtain a new general bound on the grid size of the simplicial polytope realizing a given triangulation, subexponential in a number of special cases. Formally, we prove that every plane tr ..."
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We give a new proof of Steinitz's classical theorem in the case of plane triangulations, which allows us to obtain a new general bound on the grid size of the simplicial polytope realizing a given triangulation, subexponential in a number of special cases. Formally, we prove that every plane
A Practical Bayesian Framework for Backprop Networks
 Neural Computation
, 1991
"... A quantitative and practical Bayesian framework is described for learning of mappings in feedforward networks. The framework makes possible: (1) objective comparisons between solutions using alternative network architectures ..."
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Cited by 496 (20 self)
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A quantitative and practical Bayesian framework is described for learning of mappings in feedforward networks. The framework makes possible: (1) objective comparisons between solutions using alternative network architectures
Convex Analysis
, 1970
"... In this book we aim to present, in a unified framework, a broad spectrum of mathematical theory that has grown in connection with the study of problems of optimization, equilibrium, control, and stability of linear and nonlinear systems. The title Variational Analysis reflects this breadth. For a lo ..."
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Cited by 5350 (67 self)
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progressed also to the study of socalled stationary points, critical points, and other indications of singularity that a point might have relative to its neighbors, especially in association with existence theorems for differential equations.
Why a diagram is (sometimes) worth ten thousand words
 Cognitive Science
, 1987
"... We distinguish diagrammatic from sentential paperandpencil representationsof information by developing alternative models of informationprocessing systems that are informationally equivalent and that can be characterized as sentential or diagrammatic. Sentential representations are sequential, li ..."
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Cited by 777 (2 self)
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, like the propositions in a text. Dlogrammotlc representations ore indexed by location in a plane. Diogrommatic representations also typically display information that is only implicit in sententiol representations and that therefore has to be computed, sometimes at great cost, to make it explicit
Impulses and Physiological States in Theoretical Models of Nerve Membrane
 Biophysical Journal
, 1961
"... ABSTRACT Van der Pol's equation for a relaxation oscillator is generalized by the addition of terms to produce a pair of nonlinear differential equations with either a stable singular point or a limit cycle. The resulting "BVP model " has two variables of state, representing excitabi ..."
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Cited by 496 (0 self)
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excitability and refractoriness, and qualitatively resembles Bonhoeffer's theoretical model for the iron wire model of nerve. This BVP model serves as a simple representative of a class of excitableoscillatory systems including the HodgkinHuxley (HH) model of the squid giant axon. The BVP phase plane
The information bottleneck method
 University of Illinois
, 1999
"... We define the relevant information in a signal x ∈ X as being the information that this signal provides about another signal y ∈ Y. Examples include the information that face images provide about the names of the people portrayed, or the information that speech sounds provide about the words spoken. ..."
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Cited by 545 (38 self)
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We define the relevant information in a signal x ∈ X as being the information that this signal provides about another signal y ∈ Y. Examples include the information that face images provide about the names of the people portrayed, or the information that speech sounds provide about the words spoken. Understanding the signal x requires more than just predicting y, it also requires specifying which features of X play a role in the prediction. We formalize this problem as that of finding a short code for X that preserves the maximum information about Y. That is, we squeeze the information that X provides about Y through a ‘bottleneck ’ formed by a limited set of codewords ˜X. This constrained optimization problem can be seen as a generalization of rate distortion theory in which the distortion measure d(x, ˜x) emerges from the joint statistics of X and Y. This approach yields an exact set of self consistent equations for the coding rules X → ˜ X and ˜ X → Y. Solutions to these equations can be found by a convergent re–estimation method that generalizes the Blahut–Arimoto algorithm. Our variational principle provides a surprisingly rich framework for discussing a variety of problems in signal processing and learning, as will be described in detail elsewhere. 1 1
Markov Random Field Models in Computer Vision
, 1994
"... . A variety of computer vision problems can be optimally posed as Bayesian labeling in which the solution of a problem is defined as the maximum a posteriori (MAP) probability estimate of the true labeling. The posterior probability is usually derived from a prior model and a likelihood model. The l ..."
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Cited by 515 (18 self)
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. A variety of computer vision problems can be optimally posed as Bayesian labeling in which the solution of a problem is defined as the maximum a posteriori (MAP) probability estimate of the true labeling. The posterior probability is usually derived from a prior model and a likelihood model. The latter relates to how data is observed and is problem domain dependent. The former depends on how various prior constraints are expressed. Markov Random Field Models (MRF) theory is a tool to encode contextual constraints into the prior probability. This paper presents a unified approach for MRF modeling in low and high level computer vision. The unification is made possible due to a recent advance in MRF modeling for high level object recognition. Such unification provides a systematic approach for vision modeling based on sound mathematical principles. 1 Introduction Since its beginning in early 1960's, computer vision research has been evolving from heuristic design of algorithms to syste...
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