Stable States and Sufficient Conditions for Correct

IETE Journal of Research 2003, pp 55-58 Vol49, No l, January-February,

StableStatesand SufficientConditionsfor Correct Retrievalin the BidirectionalAssociativeMemory V RAVICHANDRAN SriVenkateswara Collegeof Engineering, Departmentof Mathematicsand ComputerApplications, Pennalur,Sriperumbudur 602 105,India AND

NARAYANANSRINIVASAN Collegeof Engineering, Engineering, Sri Venkateswara Departmentof Electronicsand Communication 602 105,lndia. Pennalur,Sriperumbudur Bidirectional associative memories are used in pattern recognition applications and store the given input-output pairs. An important consideration in pattern association applications is the stability and recall accuracy of a given neural network. Some conditions for stability of library vectors using Hamming distances in a bidirectignal associative memory are discussed. We also discuss the conditions for correct retrieval of stable states in a bidirectional associative memory. We derive the conditions for correct retrieval in terms of the maximum and minimum of the correlations as well as the number of pattern pairs and the size of the pattern vectors. memory,Hammingdistance,Stablestate. Indexingterms: Bidirectionalassociative

bidirectional associative memory (BAM) usedHammingdistance.In addition,they derivedsufficient THE I developedby Bart Kosko [,2] is a heteroassociative conditionsfor correctrecall basedon the maximum of the structure; it is a generalizationof the Hopfield one layer correlations.In this paper we derive the conditions for unidirectional autoassociative memory to two layer stabilityin termsof the Hamming distance.In addition we bidirectional memory. BAM operation consists of two show that the sufficient conditions for recall are actually stages:encoding and recall. Neuronswithin a layer are not basedon the maximum of the absolutecorrelationsrather connected with each other but is connected to all the than the maximum of the correlations. neurons in the other layer. The interconnections are Haryono et al I5l suggestedthat the usage of more characterizedby weights which are set for a given inputparameters for the capacityestimatecould be beneficial. In output setof vectorsduring the encoding or learningphase. paper we extend the conditions for retrieval by using During recall, a typically noisy input vectoris presentedto this parameters, the maximum of the absolutecorrelations two the input layer (or vice versa).The correspondingoutput of the absolutecorrelationsamong well the minimum as as vectors are computed by computing a weighted sum of patterns to the length of the stored in addition stored inputs and applying a hard limiting thresholdfunction in patternpairs. of stored patterns number and the the caseof a binary or discreteBAM. The current outputs are fed back as inputs to the input layer which performs ASSOCIATIVE OF BIDIRECTIONAL STABILITY similar computationsas the output layer. This bidirectional MEMORIES operation continues until the.BAM converges.BAM is used to associatepattern vector pairs and has been used in BAM consistsof two fields of processingelementsand various applications [3J. BAM can be used as a content usesboth forward and backwardinformation flow between addressablememory and is useful for pattern recognition the two layers to recall the stored stimulus response with corruptedor noisy patterns. associations. Zhang et al l4l analyzed the BAM in terms of the matchedfiltering viewpoint and derived the conditionsfor a stablestateand correct retrieval of such stablestatesin a BAM. They derived the condition for stability basedon the effective Hamming distanceinsteadof the more commonly

The BAM stores bipolar binary row vector pairs (D)i = l, ...M, (calledlibrary vectorsof the BAM), 7Xo, Y wherey(i) . {1, -l }N and ]li) e { l, -11P. The encoding methodfor the BAM is to usethe correlationmatrix:

ll=

PaperNo 261-A;CopyrightO 2003by the IETE.

x

i=l f,3

y{ otyQ)

IETEJOURNALOF RESEARCH, Vot 49, No l. 2003 and its transpose W r as connection weight matrices betweenthe two layersof neurons.

= y'y'+

(N@y0n)r1 yr@ttytit

I i= I,i*m

The BAM beginsto evolve from the initial input X (or respectivelyl') accordingto the stateequations

M

>N-

" ,

M

\

i=l

I'= sgn(XW= sgn ( E t;g '

N . M- |

Hence

distance between XandIZdenoted bvrI(X'r)is

y{^trr= ,=#*^ lxti) ,=#*^ fi

llHtTt

=r.

I

H(x,n=; (N -YX\.

Therefore(X @t,y @)1 is a stablestatefor BAM.

Theorem 1. If a library vector (X@), l^)), satisfies the conditions

ffi3H(Y@)'vtD)s#"

(l)

| /

(Z)

Using effective Hamming distances, Zhang et al t4) obtainedthe following result.

foralli e{1, ...,M:,i*m,then (x'),y1o)isastablestate ?::::lr[rl;to'If of theBAM. Proof. We usethe signal-noisedecomposition.Notice that VI^\z= L,Xtu)y(m)t-N, l ylD ylm)l- I and y(i) y(ntr >__lX{i)y{mt Uring thesewe seethat ;. , M

yo