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896
Probabilistic independence networks for hidden Markov probability models
, 1996
"... Graphical techniques for modeling the dependencies of random variables have been explored in a variety of different areas including statistics, statistical physics, artificial intelligence, speech recognition, image processing, and genetics. Formalisms for manipulating these models have been develop ..."
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Cited by 167 (12 self)
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Graphical techniques for modeling the dependencies of random variables have been explored in a variety of different areas including statistics, statistical physics, artificial intelligence, speech recognition, image processing, and genetics. Formalisms for manipulating these models have been developed relatively independently in these research communities. In this paper we explore hidden Markov models (HMMs) and related structures within the general framework of probabilistic independence networks (PINs). The paper contains a selfcontained review of the basic principles of PINs. It is shown that the wellknown forwardbackward (FB) and Viterbi algorithms for HMMs are special cases of more general inference algorithms for arbitrary PINs. Furthermore, the existence of inference and estimation algorithms for more general graphical models provides a set of analysis tools for HMM practitioners who wish to explore a richer class of HMM structures. Examples of relatively complex models to handle sensor fusion and coarticulation in speech recognition are introduced and treated within the graphical model framework to illustrate the advantages of the general approach.
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.
Microlocal analysis and interacting quantum field theory: Renormalizability of ϕ 4
 Operator Algebras and Quantum Field Theory. Proceedings
, 1996
"... Dedicated to the memory of Professor Roberto Stroffolini Abstract. We present a perturbative construction of interacting quantum field theories on smooth globally hyperbolic (curved) spacetimes. We develop a purely local version of the StückelbergBogoliubovEpsteinGlaser method of renormalization ..."
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Cited by 92 (15 self)
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Dedicated to the memory of Professor Roberto Stroffolini Abstract. We present a perturbative construction of interacting quantum field theories on smooth globally hyperbolic (curved) spacetimes. We develop a purely local version of the StückelbergBogoliubovEpsteinGlaser method of renormalization by using techniques from microlocal analysis. Relying on recent results of Radzikowski, Köhler and the authors about a formulation of a local spectrum condition in terms of wave front sets of correlation functions of quantum fields on curved spacetimes, we construct timeordered operatorvalued products of Wick polynomials of free fields. They serve as building blocks for a local (perturbative) definition of interacting fields. Renormalization in this framework amounts to extensions of expectation values of timeordered products to all points of spacetime. The extensions are classified according to a microlocal generalization of Steinmann scaling degree corresponding to the degree of divergence in other renormalization schemes. As a result, we prove that the usual perturbative classification of interacting quantum
Enumeration of Rational Curves via Torus Actions
 in The Moduli Space of Curves, Dijkgraaf et al eds., Progress in Mathematics 129, Birkhäuser
, 1995
"... This paper contains an attempt to formulate rigorously and to check predictions in enumerative geometry of curves following from Mirror Symmetry. In a sense, we almost solved both problems. There are still certain gaps in the foundations. Nevertheless, we obtain “closed ” formulas for generating fun ..."
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Cited by 56 (0 self)
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This paper contains an attempt to formulate rigorously and to check predictions in enumerative geometry of curves following from Mirror Symmetry. In a sense, we almost solved both problems. There are still certain gaps in the foundations. Nevertheless, we obtain “closed ” formulas for generating functions in
Statistical Inference, Occam’s Razor, and Statistical Mechanics on the Space of Probability Distributions
, 1997
"... The task of parametric model selection is cast in terms of a statistical mechanics on the space of probability distributions. Using the techniques of lowtemperature expansions, I arrive at a systematic series for the Bayesian posterior probability of a model family that significantly extends known ..."
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Cited by 54 (3 self)
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The task of parametric model selection is cast in terms of a statistical mechanics on the space of probability distributions. Using the techniques of lowtemperature expansions, I arrive at a systematic series for the Bayesian posterior probability of a model family that significantly extends known results in the literature. In particular, I arrive at a precise understanding of how Occam’s razor, the principle that simpler models should be preferred until the data justify more complex models, is automatically embodied by probability theory. These results require a measure on the space of model parameters and I derive and discuss an interpretation of Jeffreys ’ prior distribution as a uniform prior over the distributions indexed by a family. Finally, I derive a theoretical index of the complexity of a parametric family relative to some true distribution that I call the razor of the model. The form of the razor immediately suggests several interesting questions in the theory of learning that can be studied using the techniques of statistical mechanics.
On motives associated to graph polynomials
 Commun. Math. Phys
"... Abstract. The appearance of multiple zeta values in anomalous dimensions and βfunctions of renormalizable quantum field theories has given evidence towards a motivic interpretation of these renormalization group functions. In this paper we start to hunt the motive, restricting our attention to a su ..."
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Cited by 50 (12 self)
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Abstract. The appearance of multiple zeta values in anomalous dimensions and βfunctions of renormalizable quantum field theories has given evidence towards a motivic interpretation of these renormalization group functions. In this paper we start to hunt the motive, restricting our attention to a subclass of graphs in four dimensional scalar field theory which give scheme independent contributions to the above functions. Calculations of Feynman integrals arising in perturbative quantum field theory [4, 5] reveal interesting patterns of zeta and multiple zeta values. Clearly, these are motivic in origin, arising from the existence of Tate mixed Hodge structures with periods given by Feynman integrals.
Topological gravity as large N topological gauge theory
 Adv. Theor. Math. Phys
, 1998
"... We consider topological closed string theories on CalabiYau manifolds which compute superpotential terms in the corresponding compactified type II effective action. In particular, near certain singularities we compare the partition function of this topological theory (the KodairaSpencer theory) to ..."
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Cited by 38 (4 self)
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We consider topological closed string theories on CalabiYau manifolds which compute superpotential terms in the corresponding compactified type II effective action. In particular, near certain singularities we compare the partition function of this topological theory (the KodairaSpencer theory) to SU(∞) ChernSimons theory on the vanishing 3cycle. We find agreement between these theories, which we check explicitly for the case of shrinking S3 and Lens spaces, at the perturbative level. Moreover, the gauge theory has nonperturbative contributions which have a natural interpretation in the Type IIB picture. We provide a heuristic explanation for this agreement as well as suggest further equivalences in other topological gravity/gauge systems. February
Worldsheet formulations of gauge theories and gravity. talk given at the 7th Marcel Grossmann Meeting Stanford
, 1994
"... The evolution operator for states of gauge theories in the graph representation (closely related to the loop representation) is formulated as a weighted sum over worldsheets interpolating between initial and final graphs. As examples, lattice SU(2) BF and YangMills theories are expressed as worldsh ..."
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Cited by 38 (7 self)
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The evolution operator for states of gauge theories in the graph representation (closely related to the loop representation) is formulated as a weighted sum over worldsheets interpolating between initial and final graphs. As examples, lattice SU(2) BF and YangMills theories are expressed as worldsheet theories, and (formal) worldsheet forms of several continuum U(1) theories are given. It is argued that the world sheet framework should be ideal for representing GR, at least euclidean GR, in 4 dimensions, because it is adapted to both the 4diffeomorphism invariance of GR, and the discreteness of 3geometry found in the loop representation quantization of the theory. However, the weighting of worldsheets in GR has not yet been found. 1
Exact Form Factors in Integrable Quantum Field Theories: the SineGordon Model
"... We provide detailed arguments on how to derive properties of generalized form factors, originally proposed by one of the authors (M.K.) and Weisz twenty years ago, solely based on the assumption of “maximal analyticity ” and the validity of the LSZ reduction formalism. These properties constitute co ..."
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Cited by 38 (18 self)
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We provide detailed arguments on how to derive properties of generalized form factors, originally proposed by one of the authors (M.K.) and Weisz twenty years ago, solely based on the assumption of “maximal analyticity ” and the validity of the LSZ reduction formalism. These properties constitute consistency equations which allow the explicit evaluation of the nparticle form factors once the scattering matrix is known. The equations give rise to a matrix RiemannHilbert problem. Exploiting the “offshell ” Bethe ansatz we propose a general formula for form factors for an odd number of particles. For the SineGordon model alias the massive Thirring model we exemplify the general solution for several operators. In particular we calculate the three particle form factor of the soliton field, carry out a consistency check against the Thirring model perturbation theory and thus confirm the general formalism. 1
Combinatorial Hopf algebras in quantum field theory I
 Reviews of Mathematical Physics
, 2005
"... This manuscript collects and expands for the most part a series of lectures on the interface between combinatorial Hopf algebra theory (CHAT) and renormalization theory, delivered by the secondnamed author in the framework of the joint mathematical physics seminar of the Universités d’Artois and Li ..."
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Cited by 35 (3 self)
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This manuscript collects and expands for the most part a series of lectures on the interface between combinatorial Hopf algebra theory (CHAT) and renormalization theory, delivered by the secondnamed author in the framework of the joint mathematical physics seminar of the Universités d’Artois and Lille 1, from late January till midFebruary 2003. The plan is as follows: Section 1 is the introduction, and Section 2 contains an elementary invitation to the subject. Sections 3–7 are devoted to the basics of Hopf algebra theory and examples, in ascending level of complexity. Section 8 contains a first, direct approach to the Faà di Bruno Hopf algebra. Section 9 gives applications of that to quantum field theory and Lagrange reversion. Section 10 rederives the Connes– Moscovici algebras. In Section 11 we turn to Hopf algebras of Feynman graphs. Then in Section 12 we give an extremely simple derivation of (the properly combinatorial part of) Zimmermann’s method, in its original diagrammatic form. In Section 13 general incidence algebras are introduced. In the next section the Faà di Bruno bialgebras