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45,384
The asymptotic number of clawfree cubic graphs
 DISCRETE MATH
, 1998
"... Let Hn be the number of clawfree cubic graphs on 2n labeled nodes. In an earlier paper we characterized clawfree cubic graphs and derived a recurrence relation for Hn. Here we determine the asymptotic behavior of this sequence: Hn (2n)! p 6 n e n 2e n=3 e (n=2)1=3 We have veri ed this formula usi ..."
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Cited by 4 (2 self)
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Let Hn be the number of clawfree cubic graphs on 2n labeled nodes. In an earlier paper we characterized clawfree cubic graphs and derived a recurrence relation for Hn. Here we determine the asymptotic behavior of this sequence: Hn (2n)! p 6 n e n 2e n=3 e (n=2)1=3 We have veri ed this formula
ClawFree Graphs  a Survey.
, 1996
"... In this paper we summarize known results on clawfree graphs. The paper is subdivided into the following chapters and sections: 1. Introduction 2. Paths, cycles, hamiltonicity a) Preliminaries b) Degree and neighborhood conditions c) Local connectivity conditions d) Further forbidden subgraph ..."
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Cited by 12 (1 self)
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In this paper we summarize known results on clawfree graphs. The paper is subdivided into the following chapters and sections: 1. Introduction 2. Paths, cycles, hamiltonicity a) Preliminaries b) Degree and neighborhood conditions c) Local connectivity conditions d) Further forbidden
Counting clawfree cubic graphs
"... Abstract. Let Hn be the number of clawfree cubic graphs on 2n labeled nodes. Combinatorial reductions are used to derive a second order, linear homogeneous differential equation with polynomial coefficients whose power series solution is the exponential generating function for {Hn}. This leads to a ..."
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Cited by 3 (1 self)
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Abstract. Let Hn be the number of clawfree cubic graphs on 2n labeled nodes. Combinatorial reductions are used to derive a second order, linear homogeneous differential equation with polynomial coefficients whose power series solution is the exponential generating function for {Hn}. This leads
Graphical models, exponential families, and variational inference
, 2008
"... The formalism of probabilistic graphical models provides a unifying framework for capturing complex dependencies among random variables, and building largescale multivariate statistical models. Graphical models have become a focus of research in many statistical, computational and mathematical fiel ..."
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Cited by 800 (26 self)
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The formalism of probabilistic graphical models provides a unifying framework for capturing complex dependencies among random variables, and building largescale multivariate statistical models. Graphical models have become a focus of research in many statistical, computational and mathematical fields, including bioinformatics, communication theory, statistical physics, combinatorial optimization, signal and image processing, information retrieval and statistical machine learning. Many problems that arise in specific instances — including the key problems of computing marginals and modes of probability distributions — are best studied in the general setting. Working with exponential family representations, and exploiting the conjugate duality between the cumulant function and the entropy for exponential families, we develop general variational representations of the problems of computing likelihoods, marginal probabilities and most probable configurations. We describe how a wide varietyof algorithms — among them sumproduct, cluster variational methods, expectationpropagation, mean field methods, maxproduct and linear programming relaxation, as well as conic programming relaxations — can all be understood in terms of exact or approximate forms of these variational representations. The variational approach provides a complementary alternative to Markov chain Monte Carlo as a general source of approximation methods for inference in largescale statistical models.
Shape Matching and Object Recognition Using Shape Contexts
 IEEE Transactions on Pattern Analysis and Machine Intelligence
, 2001
"... We present a novel approach to measuring similarity between shapes and exploit it for object recognition. In our framework, the measurement of similarity is preceded by (1) solv ing for correspondences between points on the two shapes, (2) using the correspondences to estimate an aligning transform ..."
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Cited by 1787 (21 self)
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We present a novel approach to measuring similarity between shapes and exploit it for object recognition. In our framework, the measurement of similarity is preceded by (1) solv ing for correspondences between points on the two shapes, (2) using the correspondences to estimate an aligning transform. In order to solve the correspondence problem, we attach a descriptor, the shape context, to each point. The shape context at a reference point captures the distribution of the remaining points relative to it, thus offering a globally discriminative characterization. Corresponding points on two similar shapes will have similar shape con texts, enabling us to solve for correspondences as an optimal assignment problem. Given the point correspondences, we estimate the transformation that best aligns the two shapes; reg ularized thin plate splines provide a flexible class of transformation maps for this purpose. The dissimilarity between the two shapes is computed as a sum of matching errors between corresponding points, together with a term measuring the magnitude of the aligning trans form. We treat recognition in a nearestneighbor classification framework as the problem of finding the stored prototype shape that is maximally similar to that in the image. Results are presented for silhouettes, trademarks, handwritten digits and the COIL dataset.
The pyramid match kernel: Discriminative classification with sets of image features
 IN ICCV
, 2005
"... Discriminative learning is challenging when examples are sets of features, and the sets vary in cardinality and lack any sort of meaningful ordering. Kernelbased classification methods can learn complex decision boundaries, but a kernel over unordered set inputs must somehow solve for correspondenc ..."
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Cited by 546 (29 self)
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Discriminative learning is challenging when examples are sets of features, and the sets vary in cardinality and lack any sort of meaningful ordering. Kernelbased classification methods can learn complex decision boundaries, but a kernel over unordered set inputs must somehow solve for correspondences – generally a computationally expensive task that becomes impractical for large set sizes. We present a new fast kernel function which maps unordered feature sets to multiresolution histograms and computes a weighted histogram intersection in this space. This “pyramid match” computation is linear in the number of features, and it implicitly finds correspondences based on the finest resolution histogram cell where a matched pair first appears. Since the kernel does not penalize the presence of extra features, it is robust to clutter. We show the kernel function is positivedefinite, making it valid for use in learning algorithms whose optimal solutions are guaranteed only for Mercer kernels. We demonstrate our algorithm on object recognition tasks and show it to be accurate and dramatically faster than current approaches.
Bayesian Interpolation
 Neural Computation
, 1991
"... Although Bayesian analysis has been in use since Laplace, the Bayesian method of modelcomparison has only recently been developed in depth. In this paper, the Bayesian approach to regularisation and modelcomparison is demonstrated by studying the inference problem of interpolating noisy data. T ..."
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Cited by 721 (17 self)
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Although Bayesian analysis has been in use since Laplace, the Bayesian method of modelcomparison has only recently been developed in depth. In this paper, the Bayesian approach to regularisation and modelcomparison is demonstrated by studying the inference problem of interpolating noisy data. The concepts and methods described are quite general and can be applied to many other problems. Regularising constants are set by examining their posterior probability distribution. Alternative regularisers (priors) and alternative basis sets are objectively compared by evaluating the evidence for them. `Occam's razor' is automatically embodied by this framework. The way in which Bayes infers the values of regularising constants and noise levels has an elegant interpretation in terms of the effective number of parameters determined by the data set. This framework is due to Gull and Skilling. 1 Data modelling and Occam's razor In science, a central task is to develop and compare models to a...
Mtree: An Efficient Access Method for Similarity Search in Metric Spaces
, 1997
"... A new access meth d, called Mtree, is proposed to organize and search large data sets from a generic "metric space", i.e. whE4 object proximity is only defined by a distance function satisfyingth positivity, symmetry, and triangle inequality postulates. We detail algorith[ for insertion o ..."
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Cited by 652 (38 self)
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A new access meth d, called Mtree, is proposed to organize and search large data sets from a generic "metric space", i.e. whE4 object proximity is only defined by a distance function satisfyingth positivity, symmetry, and triangle inequality postulates. We detail algorith[ for insertion of objects and split management, whF h keep th Mtree always balanced  severalheralvFV split alternatives are considered and experimentally evaluated. Algorithd for similarity (range and knearest neigh bors) queries are also described. Results from extensive experimentationwith a prototype system are reported, considering as th performance criteria th number of page I/O's and th number of distance computations. Th results demonstratethm th Mtree indeed extendsth domain of applicability beyond th traditional vector spaces, performs reasonably well inhE[94Kv#E44V[vh data spaces, and scales well in case of growing files. 1
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