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AbslracL A neural network model wilh aplimal connections trained wilh ensembles of external, discrete, noisy fields is sludied. Allowing for non-zero erron in the storage, nwel behaviour is observed which is reflected in the model's retrieval map. Improvement in the model's content addressability is determined by comparing the maximum storage level at which there is a near 100% basin of atlraction. The cases presented here have the external field applied during training, during relrieval, and during bolh with statistically equal parameters. In all three the conlent addressability is improved over the zero external field network, wilh lhe equal lraining and relrieval fields case having lhe largest improvement. However, lhe apparent dominalion of lhe retrieval over the training field suggesls this simple equality is perhaps no1 the optimal relationship. i. rntr&uctionThe key feature of statistical mechanical models of neural networks is their ability to function as associative memories. This is a two-stage process with the network first trained to store a set of memory patterns, which are then later refrieved by the neurons' update dynamics. Retrieval of the desired pattern will be successful if the system is the memory pattem's alfracfor. &pressed in this way, content addressability is merely the consequence of having finite basins of attraction.The principle aim of this work is to examine how the basin of attraction can he enlarged by the use of memal fields which are noisy representations of the memory patterns stored. Independent work has shown the beneficial effects of applying noisy external fields throughout retrieval (Engel er a1 1990) but as stated above, a network is defined in two stages and their role during the training phase should be explored. Moreover, both simulation (Gardner el a! 1989) a i d analytical (Wong and Sherrington 1990a,b) results have shown that training a network with ensembles of noisy representations also improves content addressability.For these reasons this work calculates the properties of a network trained with ensembles of noisy external fields. The retrieval dynamics under a persistent, nohy external field is then examined, and the effects of the two fields compared. By looking at the fixed-point behaviour of the dynamics, the attractor structure is revealed, and from this it is judged whether content addressability has been improved. Finally t E-mail: huyau@edinburgh.ac.uk 0305-4470/91/235639+ 12103.50 0 1991 IOP Publishing Ltd 5639 i n i t i s t d rnffir;mntlr, r l m r +n i t T h o t ir i f i t ir rtnrtorl inr;rlo tho hncin -f niimr,in..

We investigate the repulsion of random walks ( R W S ) and self-avoiding walks (SAWS) induced by ( a ) excluding a single lattice point or ( b ) excluding all points on a half-line from --OO to the origin. For SAWS, we use exact enumeration and Monte Carlo methods to study the asymptotic behaviour of the displacement away from the excluded set on three-, fourand five-dimensional hypercubic lattices. When the SAW begins one lattice site away from an excluded point along the x direction, the mean displacement after N steps, ( x N ) approaches a finite limit, at a power-law rate, as N + a . However, the distribution of projected displacements exhibits a residual asymmetry in the asymptotic limit, reflective of a long-range influence of the excluded point. This general pattern of behaviour also occurs for a purely random walk. For a SAW which starts one lattice spacing away from the axis of the excluded half-line, enumeration data suggest that the mean displacement along the axis diverges as N', with Y =: in two dimensions, but as N', with z = 0.35, in three dimensions. This unexpected behaviour appears to be corroborated by constant-fugacity Monte Carlo data. However, for the corresponding RW model, both the solution to the diffusion equation and a heuristic argument indicate that (x,.,) diverges as N " 2 in two dimensions, and as N'"/ln N in three dimensions. Large values of N are needed before indications of this asymptotic behaviour are seen numerically. Similar crossover effects may be masking the asymptotic behaviour in the SAW problem.

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