Geodesic
Stability thresholds of attractors of the Hopfield network
Izvestiya VUZ. Applied Nonlinear Dynamics, Tome 31 (2023) no. 1, pp. 75-85.

Voir la notice de l'article provenant de la source Math-Net.Ru

Purpose of the work is the detailed study of the attractors of the Hopfield network and their basins of attraction depending on the parameters of the system, the size of the network and the number of stored images. To characterize the basins of attraction we used the method of the so-called stability threshold, i.e., the minimum distance from an attractor to the boundary of its basin of attraction. For useful attractors, this value corresponds to the minimum distortion of the stored image, after which the system is unable to recognize it. In the result of the study it is shown that the dependence of the average stability threshold of useful attractors on the number of stored images can be nonmonotonic, due to which the stability of the network can improve when new images are memorized. An analysis of the stability thresholds allowed to estimate the maximum number of images that the network can store without fatal errors in their recognition. In this case, the stability threshold of useful attractors turns out to be close to the minimum possible value, that is, to unity. To conclude, calculation of the stability thresholds provides important information about the attraction basins of the network attractors.
Keywords: dynamical networks, collective dynamics, associative memory. 
@article{IVP_2023_31_1_a6,
     author = {I. A. Soloviev and V. V. Klinshov},
     title = {Stability thresholds of attractors of the {Hopfield} network},
     journal = {Izvestiya VUZ. Applied Nonlinear Dynamics},
     pages = {75--85},
     publisher = {mathdoc},
     volume = {31},
     number = {1},
     year = {2023},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/IVP_2023_31_1_a6/}
}
TY  - JOUR
AU  - I. A. Soloviev
AU  - V. V. Klinshov
TI  - Stability thresholds of attractors of the Hopfield network
JO  - Izvestiya VUZ. Applied Nonlinear Dynamics
PY  - 2023
SP  - 75
EP  - 85
VL  - 31
IS  - 1
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/IVP_2023_31_1_a6/
LA  - ru
ID  - IVP_2023_31_1_a6
ER  - 
%0 Journal Article
%A I. A. Soloviev
%A V. V. Klinshov
%T Stability thresholds of attractors of the Hopfield network
%J Izvestiya VUZ. Applied Nonlinear Dynamics
%D 2023
%P 75-85
%V 31
%N 1
%I mathdoc
%U http://geodesic.mathdoc.fr/item/IVP_2023_31_1_a6/
%G ru
%F IVP_2023_31_1_a6
I. A. Soloviev; V. V. Klinshov. Stability thresholds of attractors of the Hopfield network. Izvestiya VUZ. Applied Nonlinear Dynamics, Tome 31 (2023) no. 1, pp. 75-85. http://geodesic.mathdoc.fr/item/IVP_2023_31_1_a6/

[1] Hopfield J.J., “Neural networks and physical systems with emergent collective computational abilities”, Proc. Natl. Acad. Sci. U. S. A., 79:8 (1982), 2554–2558 | DOI | MR

[2] Hopfield J.J., “Neurons with graded response have collective computational properties like those of two-state neurons”, Proc. Natl. Acad. Sci. U. S. A, 81:10 (1984), 3088–3092. | DOI

[3] Farhat N.H., Psaltis D., Prata A., Paek E., “Optical implementation of the Hopfield model”, Applied Optics, 24:10 (1985), 1469–1475 | DOI

[4] Hoppensteadt F.C., Izhikevich E.M., “Pattern recognition via synchronization in phase-locked loop neural networks”, IEEE Transactions on Neural Networks, 11:3 (2000), 734–738 | DOI | MR

[5] Joya G., Atencia M.A., Sandoval F., “Hopfield neural networks for optimization: study of the different dynamics”, Neurocomputing, 43:1–4 (2002), 219–237 | DOI

[6] Wen U.P., Lan K.M., Shih H.S., “A review of Hopfield neural networks for solving mathematical programming problems”, European Journal of Operational Research, 198:3 (2009), 675–687 | DOI | MR

[7] McEliece R., Posner E., Rodemich E., Venkatesh S., “The capacity of the Hopfield associative memory”, IEEE Transactions on Information Theory, 33:4 (1987), 461–482 | DOI | MR

[8] Storkey A., “Increasing the capacity of a hopfield network without sacrificing functionality”, Artificial Neural Networks — ICANN'97, Lecture Notes in Computer Science, 1327, eds. Gerstner W., Germond A., Hasler M., Nicoud J.D., Springer, Berlin, Heidelberg, 1997, 451–456 | DOI

[9] Krotov D., Hopfield J.J., “Dense associative memory for pattern recognition”, Proceedings of the 30th International Conference on Neural Information Processing Systems (5–10 December 2016, Barcelona, Spain), Curran Associates Inc., New York, 2016, 1180–1188 | DOI

[10] Ramsauer H., Schäfl B., Lehner J., Seidl P., Widrich M., Adler T., Gruber L., Holzleitner M., Pavlović M., Sandve G.K., Greiff V., Kreil D., Kopp M., Klambauer G., Brandstetter J., Hochreiter S., Hopfield networks is all you need, 2020, 94 pp., arXiv: 2008.02217 https://arxiv.org/abs/2008.02217

[11] Klinshov V.V., Nekorkin V.I., Kurths J., “Stability threshold approach for complex dynamical systems”, New Journal of Physics, 18:1 (2016), 013004 | DOI | MR

[12] Menck P.J., Heitzig J., Marwan N., Kurths J., “How basin stability complements the linear-stability paradigm”, Nature Physics, 9:2 (2013), 89–92 | DOI

[13] Chakraborty A., Alam M., Dey V., Chattopadhyay A., Mukhopadhyay D., Adversarial attacks and defences: A survey, 2018, 31 pp., arXiv: 1810.00069 https://arxiv.org/abs/1810.00069

[14] Amari S.I., Maginu K., “Statistical neurodynamics of associative memory”, Neural Networks, 1:1 (1988), 63–73 | DOI