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Statistical Field Theory for Neural Networks

Aus der Reihe Lecture Notes in Physics

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Beschreibung

Produktdetails

Einband

Taschenbuch

Erscheinungsdatum

21.08.2020

Verlag

Springer

Seitenzahl

203

Maße (L/B/H)

23,5/15,5/1,3 cm

Gewicht

347 g

Auflage

1st edition 2020

Sprache

Englisch

ISBN

978-3-030-46443-1

Beschreibung

Produktdetails

Einband

Taschenbuch

Erscheinungsdatum

21.08.2020

Verlag

Springer

Seitenzahl

203

Maße (L/B/H)

23,5/15,5/1,3 cm

Gewicht

347 g

Auflage

1st edition 2020

Sprache

Englisch

ISBN

978-3-030-46443-1

Herstelleradresse

Springer-Verlag KG
Sachsenplatz 4-6
1201 Wien
AT

Email: ProductSafety@springernature.com

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  • Produktbild: Statistical Field Theory for Neural Networks
  • Produktbild: Statistical Field Theory for Neural Networks
  • I. Introduction

    II. Probabilities, moments, cumulantsA. Probabilities, observables, and momentsB. Transformation of random variablesC. CumulantsD. Connection between moments and cumulants

    III. Gaussian distribution and Wick’s theoremA. Gaussian distributionB. Moment and cumulant generating function of a GaussianC. Wick’s theoremD. Graphical representation: Feynman diagramsE. Appendix: Self-adjoint operatorsF. Appendix: Normalization of a Gaussian

    IV. Perturbation expansionA. General caseB. Special case of a Gaussian solvable theoryC. Example: Example: “phi^3 + phi^4” theoryD. External sourcesE. Cancellation of vacuum diagramsF. Equivalence of graphical rules for n-point correlation and n-th momentG. Example: “phi^3 + phi^4” theoryV. Linked cluster theoremA. General proof of the linked cluster theoremB. Dependence on j - external sources - two complimentary viewsC. Example: Connected diagrams of the “phi^3 + phi^4” theory

    VI. Functional preliminariesA. Functional derivative1. Product rule2. Chain rule3. Special case of the chain rule: Fourier transformB. Functional Taylor series

    VII. Functional formulation of stochastic differential equationsA. Onsager-Machlup path integral*B. Martin-Siggia-Rose-De Dominicis-Janssen (MSRDJ) path integralC. Moment generating functionalD. Response function in the MSRDJ formalism

    VIII. Ornstein-Uhlenbeck process: The free Gaussian theoryA. DefinitionB. Propagators in time domainC. Propagators in Fourier domain

    IX. Perturbation theory for stochastic differential equationsA. Vanishing moments of response fieldsB. Vanishing response loopsC. Feynman rules for SDEs in time domain and frequency domainD. Diagrams with more than a single external legE. Appendix: Unitary Fourier transform

    X. Dynamic mean-field theory for random networksA. Definition of the model and generating functionalB. Property of self-averagingC. Average over the quenched disorderD. Stationary statistics: Self-consistent autocorrelation of as motion of a particle in a potentialE. Transition to chaosF. Assessing chaos by a pair of identical systemsG. Schrödinger equation for the maximum Lyapunov exponentH. Condition for transition to chaos

    XI. Vertex generating functionA. Motivating example for the expansion around a non-vanishing mean valueB. Legendre transform and definition of the vertex generating function GammaC. Perturbation expansion of GammaD. Generalized one-line irreducibilityE. ExampleF. Vertex functions in the Gaussian caseG. Example: Vertex functions of the “phi^3 + phi^4”-theoryH. Appendix: Explicit cancellation until second orderI. Appendix: Convexity of WJ. Appendix: Legendre transform of a Gaussian

    XII. Application: TAP approximationInverse problem

    XIII. Expansion of cumulants into tree diagrams of vertex functionsA. Self-energy or mass operator Sigma

    XIV. Loopwise expansion of the effective action - Tree levelA. Counting the number of loopsB. Loopwise expansion of the effective action - Higher numbers of loopsC. Example: phi^3 + phi^4-theoryD. Appendix: Equivalence of loopwise expansion and infinite resummationE. Appendix: Interpretation of Gamma as effective actionF. Loopwise expansion of self-consistency equation

    XV. Loopwise expansion in the MSRDJ formalismA. Intuitive approachB. Loopwise corrections to the effective equation of motionC. Corrections to the self-energy and self-consistencyD. Self-energy correction to the full propagatorE. Self-consistent one-loopF. Appendix: Solution by Fokker-Planck equation

    XVI. Nomenclature

    Acknowledgments

    References