Bruck, Jehoshua (1990) On the Convergence Properties of the Hopfield Model. Proceedings of the IEEE, 78 (10). pp. 1579-1585. ISSN 0018-9219. doi:10.1109/5.58341. https://resolver.caltech.edu/CaltechAUTHORS:20120426-132042598
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Abstract
The main contribution of the present work is showing that the known convergence properties of the Hopfield model can be reduced to a very simple case, for which an elementary proof is provided. The convergence properties of the Hopfield model are dependent on the structure of the interconnections matrix W and the method by which the nodes are updated. Three cases are known: (1) convergence to a stable state when operating in a serial mode with symmetric W, (2) convergence to a cycle of length 2, at most, when operating in a fully parallel mode with symmetric W, and (3) convergence to a cycle of length 4 when operating in a fully parallel mode with antisymmetric W. The three known results are reviewed and it is proven that the fully parallel mode of operation is a special case of the serial model of operation. There are three more cases than can be considered using this characterization: serial mode of operation, antisymmetric W; serial mode of operation, arbitrary W; and fully parallel mode of operation, arbitrary W. By exhibiting exponential lower bounds on the length of the cycles in other cases, it is proven that the three known cases are the only interesting ones.
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| Additional Information: | © 1990 IEEE. Manuscript received July 17, 1989; revised February 21, 1990. | |||||||||
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| Issue or Number: | 10 | |||||||||
| DOI: | 10.1109/5.58341 | |||||||||
| Record Number: | CaltechAUTHORS:20120426-132042598 | |||||||||
| Persistent URL: | https://resolver.caltech.edu/CaltechAUTHORS:20120426-132042598 | |||||||||
| Official Citation: | Bruck, J.; , "On the convergence properties of the Hopfield model," Proceedings of the IEEE , vol.78, no.10, pp.1579-1585, Oct 1990 doi: 10.1109/5.58341 URL: http://ieeexplore.ieee.org/stamp/stamp.jsp?tp=&arnumber=58341&isnumber=2116 | |||||||||
| Usage Policy: | No commercial reproduction, distribution, display or performance rights in this work are provided. | |||||||||
| ID Code: | 30372 | |||||||||
| Collection: | CaltechAUTHORS | |||||||||
| Deposited By: | Ruth Sustaita | |||||||||
| Deposited On: | 26 Apr 2012 20:51 | |||||||||
| Last Modified: | 09 Nov 2021 19:46 |
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