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56
Regularization networks and support vector machines
 Advances in Computational Mathematics
, 2000
"... Regularization Networks and Support Vector Machines are techniques for solving certain problems of learning from examples – in particular the regression problem of approximating a multivariate function from sparse data. Radial Basis Functions, for example, are a special case of both regularization a ..."
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Cited by 266 (33 self)
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Regularization Networks and Support Vector Machines are techniques for solving certain problems of learning from examples – in particular the regression problem of approximating a multivariate function from sparse data. Radial Basis Functions, for example, are a special case of both regularization and Support Vector Machines. We review both formulations in the context of Vapnik’s theory of statistical learning which provides a general foundation for the learning problem, combining functional analysis and statistics. The emphasis is on regression: classification is treated as a special case.
Mean Field Theory for Sigmoid Belief Networks
 Journal of Artificial Intelligence Research
, 1996
"... We develop a mean field theory for sigmoid belief networks based on ideas from statistical mechanics. ..."
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Cited by 116 (12 self)
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We develop a mean field theory for sigmoid belief networks based on ideas from statistical mechanics.
Improving the Mean Field Approximation via the Use of Mixture Distributions
, 1998
"... Introduction Graphical models provide a formalism in which to express and manipulate conditional independence statements. Inference algorithms for graphical models exploit these independence statements, using them to compute conditional probabilities while avoiding brute force marginalization over ..."
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Cited by 38 (0 self)
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Introduction Graphical models provide a formalism in which to express and manipulate conditional independence statements. Inference algorithms for graphical models exploit these independence statements, using them to compute conditional probabilities while avoiding brute force marginalization over the joint probability table. Many inference algorithms, in particular the clustering algorithms, make explicit their usage of conditional independence by constructing a data structure that captures the essential Markov properties underlying the graph. That is, the algorithm groups interacting variables into clusters, such that the hypergraph of clusters has Markov properties that allow simple local algorithms to be employed for inference. In the best case, in which the original graph is sparse and without long cycles, the clusters are small and inference is efficient. In the worst case, such as the case of a dense graph, the clusters are large and inference is inefficient (complexity
Critical phenomena and renormalizationgroup theory, Phys. Rept
, 2002
"... We review results concerning the critical behavior of spin systems at equilibrium. We consider the Ising and the general O(N)symmetric universality class. For each of them, we review the estimates of the critical exponents, of the equation of state, of several amplitude ratios, and of the twopoint ..."
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Cited by 26 (14 self)
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We review results concerning the critical behavior of spin systems at equilibrium. We consider the Ising and the general O(N)symmetric universality class. For each of them, we review the estimates of the critical exponents, of the equation of state, of several amplitude ratios, and of the twopoint function of the order parameter. We report results in three and two dimensions. We discuss the crossover phenomena that are observed in this class of systems. In particular, we review the fieldtheoretical and numerical studies of systems with mediumrange interactions. Moreover, we consider several examples of magnetic and structural phase transitions, which are described by more complex LandauGinzburgWilson Hamiltonians, such as Ncomponent systems with cubic anisotropy, O(N)symmetric systems in the presence of quenched disorder, frustrated spin systems with noncollinear or canted order, and finally, a class of systems described by the tetragonal LandauGinzburgWilson Hamiltonian with three quartic couplings. The results for the tetragonal Hamiltonian are original, in particular we present the sixloop perturbative series for the βfunctions and the critical exponents.
A Mean Field Learning Algorithm For Unsupervised Neural Networks
, 1999
"... . We introduce a learning algorithm for unsupervised neural networks based on ideas from statistical mechanics. The algorithm is derived from a mean field approximation for large, layered sigmoid belief networks. We show how to (approximately) infer the statistics of these networks without resort to ..."
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Cited by 11 (2 self)
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. We introduce a learning algorithm for unsupervised neural networks based on ideas from statistical mechanics. The algorithm is derived from a mean field approximation for large, layered sigmoid belief networks. We show how to (approximately) infer the statistics of these networks without resort to sampling. This is done by solving the mean field equations, which relate the statistics of each unit to those of its Markov blanket. Using these statistics as target values, the weights in the network are adapted by a local delta rule. We evaluate the strengths and weaknesses of these networks for problems in statistical pattern recognition. 1. Introduction Multilayer neural networks trained by backpropagation provide a versatile framework for statistical pattern recognition. They are popular for many reasons, including the simplicity of the learning rule and the potential for discovering hidden, distributed representations of the problem space. Nevertheless, there are many issues that are...
2005a, “Loops of any size and Hamilton cycles in random scalefree networks
"... Abstract. Loops are subgraphs responsible for the multiplicity of paths going from one to another generic node in a given network. In this paper we present an analytic approach for the evaluation of the average number of loops in random scalefree networks valid at fixed number of nodes N and for an ..."
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Cited by 8 (0 self)
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Abstract. Loops are subgraphs responsible for the multiplicity of paths going from one to another generic node in a given network. In this paper we present an analytic approach for the evaluation of the average number of loops in random scalefree networks valid at fixed number of nodes N and for any length L of the loops. We bring evidence that the most frequent loop size in a scalefree network of N nodes is of the order of N like in random regular graphs while small loops are more frequent when the second moment of the degree distribution diverges. In particular, we find that finite loops of sizes larger than a critical one almost surely pass from any node, thus casting some doubts on the validity of the random tree approximation for the solution of lattice models on these graphs. Moreover we show that Hamiltonian cycles are rare in random scalefree networks and may fail to appear if the powerlaw exponent of the degree distribution is close to 2 even for minimal connectivity kmin ≥ 3.
From Useful Algorithms for Slowly Convergent Series to Physical Predictions Based on Divergent Perturbative Expansions
, 707
"... This review is focused on the borderline region of theoretical physics and mathematics. First, we describe numerical methods for the acceleration of the convergence of series. These provide a useful toolbox for theoretical physics which has hitherto not received the attention it actually deserves. T ..."
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Cited by 3 (0 self)
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This review is focused on the borderline region of theoretical physics and mathematics. First, we describe numerical methods for the acceleration of the convergence of series. These provide a useful toolbox for theoretical physics which has hitherto not received the attention it actually deserves. The unifying concept for convergence acceleration methods is that in many cases, one can reach much faster convergence than by adding a particular series term by term. In some cases, it is even possible to use a divergent input series, together with a suitable sequence transformation, for the construction of numerical methods that can be applied to the calculation of special functions. This review both aims to provide some practical guidance as well as a groundwork for the study of specialized literature. As a second topic, we review some recent developments in the field of Borel resummation, which is generally recognized as one of the most versatile methods for the summation of factorially divergent (perturbation) series. Here, the focus is on algorithms which make optimal use of all information contained in a finite set of perturbative coefficients. The unifying concept for the various aspects of the Borel method investigated here is
An Analysis of Various Elastic Net Algorithms
, 1995
"... The Elastic Net Algorithm (ENA) for solving the Traveling Salesman Problem is analyzed applying statistical mechanics. Using some general properties of the free energy function of stochastic Hopfield Neural Networks, we argue why Simic's derivation of the ENA from a Hopfield network is incorrect. ..."
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Cited by 2 (2 self)
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The Elastic Net Algorithm (ENA) for solving the Traveling Salesman Problem is analyzed applying statistical mechanics. Using some general properties of the free energy function of stochastic Hopfield Neural Networks, we argue why Simic's derivation of the ENA from a Hopfield network is incorrect. However, like the HopfieldLagrange method, the ENA may be considered a specific dynamic penalty method , where, in this case, the weights of the various penalty terms decrease during execution of the algorithm. This view on the ENA corresponds to the view resulting from the theory on `deformable templates', where the term stochastic penalty method seems to be most appropriate. Next, the ENA is analyzed both on the level of the energy function as well as on the level of the motion equations. It will be proven and shown experimentally, why a nonfeasible solution is sometimes found. It can be caused either by a too rapid lowering of the temperature parameter (which is avoidable), or...
Spindynamics simulations of the threedimensional XY model: structure factor and transport properties, Phys
 Rev. B
, 1999
"... We present extensive MonteCarlo spin dynamics simulations of the classical XY model in three dimensions on a simple cubic lattice with periodic boundary conditions. A recently developed efficient integration algorithm for the equations of motion is used, which allows a substantial improvement of st ..."
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Cited by 2 (0 self)
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We present extensive MonteCarlo spin dynamics simulations of the classical XY model in three dimensions on a simple cubic lattice with periodic boundary conditions. A recently developed efficient integration algorithm for the equations of motion is used, which allows a substantial improvement of statistics and large integration times. We find spin wave peaks in a wide range around the critical point and spin diffusion for all temperatures. At the critical point we find evidence for a violation of dynamic scaling in the sense that independent components of the dynamic structure factor S(q,ω) require different dynamic exponents in order to obtain scaling. Below the critical point we investigate the dispersion relation of the spin waves and the linewidths of S(q,ω) and find agreement with mode coupling theory. Apart from strong spin wave peaks we observe additional peaks in S(q,ω) which can be attributed to twospin wave interactions. The overall lineshapes are also discussed and compared to mode coupling predictions. Finally, we present first results for the transport coefficient D(q,ω) of the outofplane magnetization component at the critical point, which is related to the thermal conductivity of 4 He near the superfluidnormal transition.
A Statistical Superfield And Its Observable Consequences
, 1997
"... A new kind of fundamental superfield is proposed, with an Isinglike Euclidean action. Near the Planck energy it undergoes its first stage of symmetrybreaking, and the ordered phase is assumed to support specific kinds of topological defects. This picture leads to a lowenergy Lagrangian which is s ..."
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Cited by 2 (0 self)
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A new kind of fundamental superfield is proposed, with an Isinglike Euclidean action. Near the Planck energy it undergoes its first stage of symmetrybreaking, and the ordered phase is assumed to support specific kinds of topological defects. This picture leads to a lowenergy Lagrangian which is similar to that of standard physics, but there are interesting and observable di#erences. For example, the cosmological constant vanishes, fermions have an extra coupling to gravity, the gravitational interaction of Wbosons is modified, and Higgs bosons have an unconventional equation of motion. email: allen@phys.tamu.edu tel.: (409) 8454341 fax: (409) 8452590 International Journal of Modern Physics A, Vol. 12, No. 13 (1997) 23852412 CTPTAMU15/96 1 1 Introduction The terms "superfield" and "supersymmetry" are ordinarily used in a context which presupposes local Lorentz invariance. 13 It is far from clear, however, that Lorentz invariance is still valid near the Planck scale, fift...