## Self-Averaging In A Class Of Generalized Hopfield Models (1994)

Venue: | J. Phys. A |

Citations: | 4 - 2 self |

### BibTeX

@ARTICLE{Bovier94self-averagingin,

author = {Anton Bovier},

title = {Self-Averaging In A Class Of Generalized Hopfield Models},

journal = {J. Phys. A},

year = {1994},

volume = {27},

pages = {7069--7077}

}

### Years of Citing Articles

### OpenURL

### Abstract

: We prove the almost sure convergence to zero of the fluctuations of the free energy, resp. local free energies, in a class of disordered mean-field spin systems that generalize the Hopfield model in two ways: 1) Multi-spin interactions are permitted and 2) the random variables ¸ ¯ i describing the `patterns' can have arbitrary distributions with mean zero and finite 4+ ffl-th moments. The number of patterns, M , is allowed to be an arbitrary multiple of the systemsize. This generalizes a previous result of Bovier, Gayrard, and Picco [BGP3] for the standard Hopfield model, and improves a result of Feng and Tirozzi [FT] that required M to be a finite constant. Note that the convergence of the mean of the free energy is not proven. Keywords: Hopfield model, neural networks, self-averaging, law of large numbers. PACS Classification Numbers: 75.10.Nr, 64.60.Cn, 89.70.+c # Work partially supported by the Commission of the European Communities under contract CHRX-CT93-0411 1 e-mail: b...

### Citations

1780 | Neural networks and physical systems with emergent collective computation abilities - Hopfield - 1982 |

36 | Necessary and sufficient conditions for the almost sure convergence of the largest eigenvalue of a Wigner matrix - Bai, Yin - 1988 |

31 |
M.V.: Absence of self-averaging of the order parameter in the SherringtonKirkpatrick model
- Pastur, Shcherbina
- 1991
(Show Context)
Citation Context ... as it contains a parameter, the number M of stored patterns as a function of the size of the system, N which can be adjusted to alter the properties of the model. In a paper of Pastur and Shcherbina =-=[PS]-=-, it was observed that the variance of the free energy of a finite system of size N in the SK-model tends to zero like 1=N , implying the convergence to zero in probability of the difference between t... |

24 |
Machine learning using a higher order correlation network. Physica D
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(Show Context)
Citation Context ...=1 �� �� i 1 : : : �� �� i r oe i 1 : : : oe i r (1:1) Here rs2 is some chosen integer. The case r = 2 corresponds to the usual Hopfield model, and models with general r were introduce=-=d by Lee et al. [Lee]-=- and Peretto and Niez [PN]. Feng and Tirozzi [FT] also studied these models, but removed the terms in the sum where two or more indices 18=november=1995; 12:19scoincide, which actually amounts to addi... |

23 |
The free energy of the class of Hopfield models
- Shcherbina, Tirozzi
- 1993
(Show Context)
Citation Context ...ro like 1=N , implying the convergence to zero in probability of the difference between the free energy and its mean. This result was later generalized to the Hopfield model by Shcherbina and Tirozzi =-=[ST] unde-=-r the assumption that the ratio ff = M=N remains bounded as N " 1. Further results of this type can be found in an interesting paper by Pastur, Shcherbina and Tirozzi [PST]. Self-averaging proper... |

22 | P: Gibbs states of the Hopfield model in the regime of perfeect memory, Probab. Theory. Related Fields 100, no 3
- Bovier, Gayrard, et al.
- 1994
(Show Context)
Citation Context ...M N , fl = ffl 4(4+ffl) . Remark: The proof of Lemma 3 is in fact an adaptation of the the truncation idea in [BY] and fairly standard estimates on the traces of powers of A, as in [BY] (but see also =-=[BGP1]). We wi-=-ll therefore not give the details of the proof of Lemma 3, but only mention that the second term is a bound on the probability that any of the �� �� i exceeds the value p N ffi, while the firs... |

17 | Gibbs states of the Hopfield model with extensively many patterns
- Bovier, Gayrard, et al.
- 1995
(Show Context)
Citation Context ...ons with mean zero and finite 4+ ffl-th moments. The number of patterns, M , is allowed to be an arbitrary multiple of the systemsize. This generalizes a previous result of Bovier, Gayrard, and Picco =-=[BGP3]-=- for the standard Hopfield model, and improves a result of Feng and Tirozzi [FT] that required M to be a finite constant. Note that the convergence of the mean of the free energy is not proven. Keywor... |

16 | Exactly soluble model of a spin glass - Pastur, Figotin - 1977 |

15 |
Exponential inequalities for sums of random vectors
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Citation Context ...e may introduce the event A j fkAks2(1 + p ff) 2 g ae F (2:1) and the corresponding trace-sigma algebras ~ F k j F k " A. This allows to introduce the the corresponding martingale difference sequ=-=ence [Yu]-=- ~ f (k) N (fi) j IE h f N (fi) fi fi ~ F k i \Gamma IE h f N (fi) fi fi ~ F k+1 i (2:2) Notice that we have the identity ~ f N (fi) \Gamma IE ~ f N (fi) = N X k=1 ~ f (k) N (fi)IP [A] (2:3) by the de... |

14 | Large deviation principles for the Hopfield model and the Kac-Hopfield model”, Prob - Bovier, Gayrard, et al. - 1995 |

14 |
The replica symmetric solution without the replica trick for the Hopfield
- Pastur, Shcherbina, et al.
- 1994
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Citation Context ... Shcherbina and Tirozzi [ST] under the assumption that the ratio ff = M=N remains bounded as N " 1. Further results of this type can be found in an interesting paper by Pastur, Shcherbina and Tir=-=ozzi [PST]-=-. Self-averaging properties of the large deviation rate function as a function of the macroscopic parameters of the model (the so called `overlap-parameters', see below) were used crucially in two pap... |

13 |
Rigorous results on the thermodynamics of the dilute Hopfield model
- Bovier, Gayrard
- 1993
(Show Context)
Citation Context ...mploy this term, because the convergence of IEFN (fi) to a limit is in general not proven. In the standard Hopfield model this was proven under the assumption lim N"1 M(N) N = 0 by Koch [K] (see =-=also [BG]-=-). We conclude the introduction by giving the proof of Theorem 2, assuming Theorem 1. 18=november=1995; 12:19sProof: (of Theorem 2) Obviously, FN (fi) = 1 N ~ f n (fi) + FN (fi)1I fkA(N)k?aeg (1:13) B... |

12 | A free energy bound for the Hopfield model
- Koch
- 1993
(Show Context)
Citation Context ...reluctant to employ this term, because the convergence of IEFN (fi) to a limit is in general not proven. In the standard Hopfield model this was proven under the assumption lim N"1 M(N) N = 0 by =-=Koch [K]-=- (see also [BG]). We conclude the introduction by giving the proof of Theorem 2, assuming Theorem 1. 18=november=1995; 12:19sProof: (of Theorem 2) Obviously, FN (fi) = 1 N ~ f n (fi) + FN (fi)1I fkA(N... |

12 |
Eigenvalue distribution in some ensembles of random matrices
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- 1967
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Citation Context ...j 1 N M X ��=1 �� �� i �� �� j (1:7) Properties of the maximal eigenvalues of this matrix will be crucial for us. The eigenvalue distribution of this matrix was first analyzed by M=-=archenko and Pastur [MP]-=-. Girko [Gi] proved that under the hypothesis of Theorem 1, the maximal eigenvalue of A(N) converges to (1 + ff) 2 in probability. Adding the ideas used by Bai and Yin [BY] one can easily show that th... |

9 |
Long term memory storage capacity of multiconnected neural nets
- Peretto, Niez
- 1986
(Show Context)
Citation Context ...r oe i 1 : : : oe i r (1:1) Here rs2 is some chosen integer. The case r = 2 corresponds to the usual Hopfield model, and models with general r were introduced by Lee et al. [Lee] and Peretto and Niez =-=[PN]-=-. Feng and Tirozzi [FT] also studied these models, but removed the terms in the sum where two or more indices 18=november=1995; 12:19scoincide, which actually amounts to adding a term of the order of ... |

2 |
Limit theorems for maximal and minimal eigenvalues of random matrices
- Girko
- 1989
(Show Context)
Citation Context ...=1 �� �� i �� �� j (1:7) Properties of the maximal eigenvalues of this matrix will be crucial for us. The eigenvalue distribution of this matrix was first analyzed by Marchenko and Pas=-=tur [MP]. Girko [Gi]-=- proved that under the hypothesis of Theorem 1, the maximal eigenvalue of A(N) converges to (1 + ff) 2 in probability. Adding the ideas used by Bai and Yin [BY] one can easily show that this convergen... |

1 |
The SLLN for the free energy in the Hopfield type model and Little model with finite number of patterns", preprint
- Feng, Tirozzi
- 1994
(Show Context)
Citation Context ...lowed to be an arbitrary multiple of the systemsize. This generalizes a previous result of Bovier, Gayrard, and Picco [BGP3] for the standard Hopfield model, and improves a result of Feng and Tirozzi =-=[FT]-=- that required M to be a finite constant. Note that the convergence of the mean of the free energy is not proven. Keywords: Hopfield model, neural networks, self-averaging, law of large numbers. PACS ... |

1 |
Solvable model of a spin glass", Phys
- Sherrington, Kirkpatrick
- 1972
(Show Context)
Citation Context ...ntroduction Over the last years some interesting properties of `self-averaging' have been observed in two classes of `spin-glass' type models of the mean field type, the Sherrington-Kirkpatrick model =-=[SK]-=- and the Hopfield model [FP, Ho]. The latter, largely used in the context of neural networks, is maybe particularly interesting, as it contains a parameter, the number M of stored patterns as a functi... |