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  • 1
    Digitale Medien
    Digitale Medien
    The retrieval phase of the Hopfield model: A rigorous analysis of the overlap distribution* (1997)
    Springer
    Probability theory and related fields 107 (1997), S. 61-98 
    Details
    ISSN: 1432-2064
    Schlagwort(e): Mathematics Subject Classification (1991): 82B44 ; 60K35 ; 82C32
    Quelle: Springer Online Journal Archives 1860-2000
    Thema: Mathematik
    Notizen: Summary. Standard large deviation estimates or the use of the Hubbard–Stratonovich transformation reduce the analysis of the distribution of the overlap parameters essentially to that of an explicitly known random function Φ N,β on ℝ M . In this article we present a rather careful study of the structure of the minima of this random function related to the retrieval of the stored patterns. We denote by m * (β ) the modulus of the spontaneous magnetization in the Curie–Weiss model and by α the ratio between the number of the stored patterns and the system size. We show that there exist strictly positive numbers 0 〈 γ a 〈 γ c such that (1) If √α≦γ a (m * (β )) 2 , then the absolute minima of Φ are located within small balls around the points ± m * e μ , where e μ denotes the μ-th unit vector while (2) if √α≦γ c (m * (β )) 2 at least a local minimum surrounded by extensive energy barriers exists near these points. The random location of these minima is given within precise bounds. These are used to prove sharp estimates on the support of the Gibbs measures.
    Materialart: Digitale Medien
    URL: http://dx.doi.org/10.1007/s004400050077
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  • 2
    Digitale Medien
    Digitale Medien
    Gibbs states of the Hopfield model in the regime of perfect memory (1994)
    Springer
    Probability theory and related fields 100 (1994), S. 329-363 
    Details
    ISSN: 1432-2064
    Schlagwort(e): 60K35 ; 82B44 ; 82C32
    Quelle: Springer Online Journal Archives 1860-2000
    Thema: Mathematik
    Notizen: Summary We study the thermodynamic properties of the Hopfield model of an autoassociative memory. IfN denotes the number of neurons andM (N) the number of stored patterns, we prove the following results: IfM/N↓ 0 asN↑ ∞, then there exists an infinite number of infinite volume Gibbs measures for all temperaturesT〈1 concentrated on spin configurations that have overlap with exactly one specific pattern. Moreover, the measures induced on the overlap parameters are Dirac measures concentrated on a single point and the Gibbs measures on spin configurations are products of Bernoulli measures. IfM/N → α, asN↓∞ for α small enough, we show that for temperaturesT smaller than someT(α)〈1, the induced measures can have support only on a disjoint union of balls around the previous points, but we cannot construct the infinite volume measures through convergent sequences of measures.
    Materialart: Digitale Medien
    URL: http://dx.doi.org/10.1007/BF01193704
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  • 3
    Digitale Medien
    Digitale Medien
    Large deviation principles for the Hopfield model and the Kac-Hopfield model (1995)
    Springer
    Probability theory and related fields 101 (1995), S. 511-546 
    Details
    ISSN: 1432-2064
    Schlagwort(e): 60K35 ; 82B44 ; 82C32
    Quelle: Springer Online Journal Archives 1860-2000
    Thema: Mathematik
    Notizen: Summary We study the Kac version of the Hopfield model and prove a Lebowitz-Penrose theorem for the distribution of the overlap parameters. At the same time, we prove a large deviation principle for the standard Hopfield model with infinitely many patterns.
    Materialart: Digitale Medien
    URL: http://dx.doi.org/10.1007/BF01202783
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  • 4
    Digitale Medien
    Digitale Medien
    Metastates in the Hopfield Model in the Replica Symmetric Regime (1998)
    Springer
    Mathematical physics, analysis and geometry 1 (1998), S. 107-144 
    Details
    ISSN: 1572-9656
    Schlagwort(e): Hopfield model ; neural networks ; metastates ; replica symmetry ; Brascamp–Lieb inequalities
    Quelle: Springer Online Journal Archives 1860-2000
    Thema: Mathematik , Physik
    Notizen: Abstract We study the finite dimensional marginals of the Gibbs measure in the Hopfield model at low temperature when the number of patterns, M, is proportional to the volume with a sufficiently small proportionality constant α 〉 0. It is shown that even when a single pattern is selected (by a magnetic field or by conditioning), the marginals do not converge almost surely, but only in law. The corresponding limiting law is constructed explicitly. We fit our result in the recently proposed language of ‘metastates’ which we discuss some length. As a byproduct, in a certain regime of the parameters α and β (the inverse temperature), we also give a simple proof of Talagrand’s recent result that the replica symmetric solution found by Amit, Gutfreund, and Sompolinsky can be rigorously justified.
    Materialart: Digitale Medien
    URL: http://dx.doi.org/10.1023/A:1009764607660
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  • 5
    Digitale Medien
    Digitale Medien
    Distribution of Overlap Profiles in the One-Dimensional Kac--Hopfield Model (1997)
    Springer
    Communications in mathematical physics 186 (1997), S. 323-379 
    Details
    ISSN: 1432-0916
    Quelle: Springer Online Journal Archives 1860-2000
    Thema: Mathematik , Physik
    Notizen: Abstract: We study a one-dimensional version of the Hopfield model with long, but finite range interactions below the critical temperature. In the thermodynamic limit we obtain large deviation estimates for the distribution of the “local” overlaps, the range of the interaction, , being the large parameter. We show in particular that the local overlaps in a typical Gibbs configuration are constant and equal to one of the mean-field equilibrium values on a scale . We also give estimates on the size of typical “jumps”, i.e. the regions where transitions from one equilibrium value to another take place. Contrary to the situation in the ferromagnetic Kac-model, the structure of the profiles is found to be governed by the quenched disorder rather than by entropy.
    Materialart: Digitale Medien
    URL: http://dx.doi.org/10.1007/s002200050112
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  • 6
    Digitale Medien
    Digitale Medien
    Efferent projections from the retrochiasmatic area to the median eminence and to the pars nervosa of the hypophysis with special reference to the A15 dopaminergic cell group in the sheep (1995)
    Springer
    Cell & tissue research 281 (1995), S. 561-567 
    Details
    ISSN: 1432-0878
    Schlagwort(e): Anterograde tracers ; Immunohistochemistry ; Tyrosine hydroxylase ; A15 dopaminergic group ; Retrochiasmatic area ; Prolactin secretion ; Sheep
    Quelle: Springer Online Journal Archives 1860-2000
    Thema: Biologie , Medizin
    Notizen: Abstract Anterograde tracers, viz. Phaseolus vulgaris leucoagglutinin and fluorescein dextran, were used in conjunction with tyrosine hydroxylase immunohisto-chemistry to study the projections of the A15 dopaminergic cell group towards the median eminence and pituitary in sheep. After injection of the tracers in the retrochiasmatic area, which contains the cell group A15, fibres containing anterograde tracer were observed in the internal zone of the median eminence and in the pars nervosa of the pituitary. Numerous tyrosine hydroxylase immunoreactive fibers were present in the external zone of the median eminence and in the pars intermedia and the pars nervosa of the pituitary, with characteristic patterns of organisation in each area. Most tyrosine hydroxylase-immunoreactive fibres containing fluorescein dextran were located in the pars nervosa, whereas only a few were observed in the internal zone of the median eminence. It was concluded that at least part of the dopaminergic innervation of the pars nervosa originated from the A15 group. These results provide morphological evidence for (1) the role of dopaminergic neurons of the A15 cell group in the seasonal control of prolactin secretion via the release of dopamine in the pars nervosa, and (2) putative physiological interactions between dopamine and the secretion of neurohypophysial hormones in sheep.
    Materialart: Digitale Medien
    URL: http://dx.doi.org/10.1007/BF00417874
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  • 7
    Digitale Medien
    Digitale Medien
    Efferent projections from the retrochiasmatic area to the median eminence and to the pars nervosa of the hypophysis with special reference to the A15 dopaminergic cell group in the sheep (1995)
    Springer
    Cell & tissue research 281 (1995), S. 561-567 
    Details
    ISSN: 1432-0878
    Schlagwort(e): Key words: Anterograde tracers ; Immunohistochemistry ; Tyrosine hydroxylase ; A15 dopaminergic group ; Retrochiasmatic area ; Prolactin secretion ; Sheep
    Quelle: Springer Online Journal Archives 1860-2000
    Thema: Biologie , Medizin
    Notizen: Abstract. Anterograde tracers, viz. Phaseolus vulgaris leucoagglutinin and fluorescein dextran, were used in conjunction with tyrosine hydroxylase immunohistochemistry to study the projections of the A15 dopaminergic cell group towards the median eminence and pituitary in sheep. After injection of the tracers in the retrochiasmatic area, which contains the cell group A15, fibres containing anterograde tracer were observed in the internal zone of the median eminence and in the pars nervosa of the pituitary. Numerous tyrosine hydroxylase immunoreactive fibres were present in the external zone of the median eminence and in the pars intermedia and the pars nervosa of the pituitary, with characteristic patterns of organisation in each area. Most tyrosine hydroxylase-immunoreactive fibres containing fluorescein dextran were located in the pars nervosa, whereas only a few were observed in the internal zone of the median eminence. It was concluded that at least part of the dopaminergic innervation of the pars nervosa originated from the A15 group. These results provide morphological evidence for (1) the role of dopaminergic neurons of the A15 cell group in the seasonal control of prolactin secretion via the release of dopamine in the pars nervosa, and (2) putative physiological interactions between dopamine and the secretion of neurohypophysial hormones in sheep.
    Materialart: Digitale Medien
    URL: http://dx.doi.org/10.1007/s004410050452
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  • 8
    Digitale Medien
    Digitale Medien
    Rigorous bounds on the storage capacity of the dilute Hopfield model (1992)
    Springer
    Journal of statistical physics 69 (1992), S. 597-627 
    Details
    ISSN: 1572-9613
    Schlagwort(e): Hopfield model ; bond dilution ; memory capacity
    Quelle: Springer Online Journal Archives 1860-2000
    Thema: Physik
    Notizen: Abstract We study a neural network model consisting ofN neurons where a dendritic connection between each pair of neurons exists with probabilityp and is absent with probability 1-p. For the Hopfield Hamiltonian on such a network, we prove that ifp ⩾ c[(lnN)/N]1/2, the model can store at leastm=α cpN patterns, whereα c ≈ 0.027 ifc ⩾∼ 3 and decreases proportional to 1/(−lnc) forc small. This generalizes the results of Newman for the standard Hopfield model.
    Materialart: Digitale Medien
    URL: http://dx.doi.org/10.1007/BF01050427
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  • 9
    Digitale Medien
    Digitale Medien
    Rigorous results on the thermodynamics of the dilute Hopfield model (1993)
    Springer
    Journal of statistical physics 72 (1993), S. 79-112 
    Details
    ISSN: 1572-9613
    Schlagwort(e): Neural networks ; Hopfield model ; random graphs ; mean-field theory
    Quelle: Springer Online Journal Archives 1860-2000
    Thema: Physik
    Notizen: Abstract We study the Hopfield model of an autoassociative memory on a random graph onN vertices where the probability of two vertices being joined by a link isp(N). Assuming thatp(N) goes to zero more slowly thanO(1/N), we prove the following results: (1) If the number of stored patternsm(N) is small enough such thatm(N)/Np(N) ↓0, asN↑∞, then the free energy of this model converges, upon proper rescaling, to that of the standard Curie-Weiss model, for almost all choices of the random graph and the random patterns. (2) If in additionm(N) 〈 ln N/ln 2, we prove that there exists, forT〈 1, a Gibbs measure associated to each original pattern, whereas for higher temperatures the Gibbs measure is unique. The basic technical result in the proofs is a uniform bound on the difference between the Hamiltonian on a random graph and its mean value.
    Materialart: Digitale Medien
    URL: http://dx.doi.org/10.1007/BF01048041
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  • 10
    Digitale Medien
    Digitale Medien
    The thermodynamics of the Curie-Weiss model with random couplings (1993)
    Springer
    Journal of statistical physics 72 (1993), S. 643-664 
    Details
    ISSN: 1572-9613
    Schlagwort(e): Curie-Weiss model ; random graphs ; disordered magnets ; mean-field theory
    Quelle: Springer Online Journal Archives 1860-2000
    Thema: Physik
    Notizen: Abstract We study the Curie-Weiss version of an Ising spin system with random, positively biased couplings. In particular, the case where the couplingsɛ ij take the values one with probabilityp and zero with probability 1 −p, which describes the Ising model on a random graph, is considered. We prove that ifp is allowed to decrease with the system sizeN in such a way thatNp(N) ↑ ∞ asN ↑ ∞, then the free energy converges (after trivial rescaling) to that of the standard Curie-Weiss model, almost surely. Similarly, the induced measures on the mean magnetizations converge to those of the Curie-Weiss model. Generalizations of this result to a wide class of distributions are detailed.
    Materialart: Digitale Medien
    URL: http://dx.doi.org/10.1007/BF01048027
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