Past exam of the mathematics course of the University of Cambridge / 2007 / iii / Paper 73 / 3 / d

A standard binary Hopfield network contains $M$ recurrently connected units with activities $x_i\in \{-1,+1\}$. Choose symmetric weights $w_{ij}=w_{ji}$, omit self-couplings by setting $w_{ii}=0$, and use zero thresholds for the energy in the next part. To store desired binary patterns ${\xi }^{\mu }$, a simple choice is Hebbian memory weights for a Hopfield network:Correlated activity strengthens connections between units agreeing across the training patterns and makes disagreement contribute negatively. This creates attraction toward the stored configurations when cross-talk is sufficiently small. For one stored pattern, its local field is $(M-1){\xi }_i/M$, so every bit is stable for $M>1$. With multiple patterns, the other patterns add cross-talk; the rule does not guarantee exact retrieval for an arbitrary collection.

Start retrieval from a partial or corrupted cue and update one unit at a time using its current fieldA cyclic or random fair asynchronous schedule lets the network relax toward a fixed point, completing a memory when the cue lies in its basin. This is content-addressable retrieval: the initial pattern selects the memory without an explicit address. Nonzero thresholds can be included in a more general model, but their corresponding linear terms must then be included in the energy.