Sharp bounds on perfect retrieval in the Hopfield model
- Bovier, Anton
2010 Mathematics Subject Classification
- 82B44 60K35 82C32
- Hopfield model, storage capacity, gradient dynamic, sequential dynamic
We prove a sharp upper bound on the number of patterns that can be stored in the Hopfield model if the stored patterns are required to be fixed points of the gradient dynamic. We also show corresponding bounds on the one-step convergence of the sequential gradient dynamic. The bounds coincide with the known lower bounds and confirm the heuristic expectations. The proof is based on a crucial idea of Loukianova [L] to use the negative association properties of some random variables arising in the analysis.
- J. Appl. Probab., 36 (1999), pp. 941-950.