{"title": "Bidirectional Retrieval from Associative Memory", "book": "Advances in Neural Information Processing Systems", "page_first": 675, "page_last": 681, "abstract": "", "full_text": "Bidirectional  Retrieval  from  Associative \n\nMemory \n\nFriedrich T.  Sommer and Gunther  Palm \nDepartment of Neural Information Processing \n\nUniversity of Ulm,  89069 Ulm,  Germany \n{sommer,palm}~informatik.uni-ulm.de \n\nAbstract \n\nSimilarity based fault  tolerant retrieval in neural associative mem(cid:173)\nories  (N AM)  has  not  lead  to  wiedespread  applications.  A  draw(cid:173)\nback  of  the  efficient  Willshaw  model  for  sparse  patterns  [Ste61, \nWBLH69],  is  that  the high  asymptotic information  capacity is  of \nlittle  practical  use  because  of  high  cross  talk  noise  arising  in  the \nretrieval for finite  sizes.  Here a new bidirectional iterative retrieval \nmethod for  the Willshaw model is presented, called crosswise bidi(cid:173)\nrectional (CB)  retrieval, providing enhanced performance.  We  dis(cid:173)\ncuss its asymptotic capacity limit,  analyze the first  step, and com(cid:173)\npare it in experiments with the Willshaw model.  Applying the very \nefficient  CB  memory model either in information retrieval systems \nor  as  a  functional  model  for  reciprocal  cortico-cortical  pathways \nrequires  more  than  robustness against  random  noise  in  the input: \nOur experiments show also the segmentation ability of CB-retrieval \nwith  addresses  containing  the  superposition  of  pattens,  provided \neven at high memory load. \n\n1 \n\nINTRODUCTION \n\nFrom  a  technical  point  of  view  neural  associative  memories  (N AM)  provide  data \nstorage  and  retrieval.  Neural  models  naturally  imply  parallel  implementation  of \nstorage  and  retrieval  algorithms  by  the  correspondence  to  synaptic  modification \nand  neural  activation.  With  distributed  coding  of  the  data  the  recall  in  N AM \nmodels is fault  tolerant:  It is robust against noise or superposition in the addresses \nand against local damage in the synaptic weight matrix.  As  biological models N AM \n\n\f676 \n\nF.  T.  Sommer and G.  Palm \n\nhave  been  proposed  as  general  working  schemes  of networks  of pyramidal  cells  in \nmany places of the cortex. \n\nAn  important  property  of  a  NAM  model  is  its  information  capacity,  measuring \nhow  efficient  the synaptic weights are used.  In  the early sixties  Steinbuch realized \nunder the name  \"Lernmatrix\"  a memory model  with binary synapses which is  now \nknown  as  Wills haw  model  [Ste6I,  WBLH69].  The  great  variety  of NAM  models \nproposed since  then, many  triggered by Hopfield's  work  [Hop82],  do  not reach  the \nhigh asymptotic information capacity of the Willshaw model. \n\nFor  finite  network  size,  the  Willshaw  model  does  not  optimally  retrieve  the \nstored  information,  since  the  inner  product  between  matrix  colum  and  input \npattern  determines  the  activity  for  each  output  neuron  independently.  For  au(cid:173)\ntoassociative  pattern  completion  iterative  retrieval  can  reduce  cross  talk  noise \n[GM76,  GR92,  PS92,  SSP96].  A simple bidirectional iteration - as in bidirectional \nassociative memory  (BAM)  [Kos87]  - can,  however,  not improve heteroassociative \npattern mapping.  For this  task we  propose  CB-retrieval where  each  retrieval step \nforms the resulting activity pattern in an autoassociative process that uses the con(cid:173)\nnectivity matrix twice before thresholding, thereby exploiting the stored information \nmore efficiently. \n\n2  WILLSHAW MODEL  AND  CB  EXTENSION \n\nHere pattern mapping tasks  XV  -+  yV  are considered for  a set of  memory  patterns: \n{(XV,yV):  XV  E  {O,I}n,yv  E {o,I}m,v = I, ... ,M}.  The  number of I-components \nin  a  pattern  is  called  pattern  activity.  The  Willshaw  model  works  efficiently,  if \nthe  memories  are  sparse,  i.e.,  if  the  memory  patterns  have  the  same  activities: \nIxvi  = 2:~=I xi = a,lyvl  = 2::1 Yi  = b V v  with  a \u00ab  nand b \u00abm.  During \nlearning the set  of memory patterns is  transformed to the weight matrix by \n\nCij  =  min(I, L xiv}) =  supxiy'j\u00b7 \n\nV \n\nV \n\nFor a  given initial pattern XJ1.  the retrieval yields the output pattern yJ1.  by forming \nin each neuron the dendritic sum [CxJ1.]j  = 2:i Ci/if and by calculating the activity \nvalue by  threshold comparison \n\nyj =  H([CxJ1.j j - 9)  Vj, \n\n(1) \n\nwith the global threshold  value 9 and H(x)  denoting the Heaviside function. \nFor finite sizes and with high memory load,  Le.,  0\u00ab  PI  := Prob[Cij =  1] \u00ab  0.5), \nthe Willshaw model provides no tolerance with respect to errors in the address, see \nFig. 1 and 2.  A bidirectional iteration of standard simple retrieval (1), as proposed in \nBAM models [Kos87], can therefore be ruled out for further retrieval error reduction \n[SP97j.  In the energy function of the Willshaw BAM \n\nE(x,y) = - LCijXiYj + 8' LXi + 8  LYj \n\nij \n\ni \n\nj \n\nwe  now indroduce a factor  accounting for  the magnitudes of dendritic potentials at \nacti vated neurons \n\n(2) \n\n\fBidirectional Retrieval from Associative Memory \n\nDifferentiating the energy function  (2)  yields  the gradient descent  equations \n\nyr W  =  H( [CxU  + L 'LCijCikXi Yk  - 8  ) \n\nX~ew  =  H( [CT y);  + L \"LPiiCljYi Xl  - 8' ) \n\nk \n\ni \n\n\" ' -v - - - '  \n\n=:Wjk \n\nI \n\ni ---------\n\n=:wfr \n\n677 \n\n(3) \n\n(4) \n\nAs  new  terms  in  (3)  and  (4)  sums  over  pattern  components  weighted  with  the \nquantities wjk  and wft  occur.  wjk  is  the overlap between the matrix columns j  and \nk  conditioned  by  the pattern  X,  which  we  call  a  conditioned  link between  y-units. \nRestriction  on  the  conditioned  link  terms  yields  a  new  iterative  retrieval  scheme \nwhich  we  denote as  crosswise  bidirectional  (eB)  retrieval \n\ny(r+ I)i  =  H( 'L Cij[CT y(r-I))i - 8) \nH( L Cij[Cx(r-I))j - 8') \n\niEx(r) \n\niEy(r) \n\n(5) \n\n(6) \n\nFor r  =  0 pattern y(r:-I)  has to be replaced by  H([Cx(O)]  - 0),  for  r  > 2 Boolean \nANDing with  results from  timestep r  - 1 can be applied  which  has been shown to \nimprove iterative retrieval in the Willshaw model for  autoassociation [SSP96]. \n\n3  MODEL EVALUATION \n\nTwo  possible  retrieval error types  can be distinguished:  a  \"miss\"  error converts a \nI-entry in  Y~ to '0'  and a  \"add\"  error does the opposite. \n\n]. \n\" \n2. \n\" \n,. \n\n35 \n30 \n25 \n20 \n15 \n10 \n5 \n0 \n\n2. \n\n]. \n\n40 \n\nsimple r.  add error  ... . . \n\nC8-r. add error  -\nCB-r. miss error  ..... \n\n5 \n\n10 \n\n15 \n\n20 \n\n25 \n\n30 \n\nFigure  1:  Mean  retrieval  error  rates  for  n  =  2000,  M  =  15000,  a  =  b  =  10 \ncorresponding  to  a  memory  load  of  H  = 0.3.  The  x-axes  display  the  address \nactivity:  lilLl  =  10 corresponds to a  errorfree learning  pattern, lower  activities are \ndue to miss  errors, higher  activities due to add errors.  Left:  Theory - Add  errors \nfor  simple  retrieval,  eq.  (7)  (upper  curve)  and  lower  bound  for  the  first  step  of \nCB-retrieval, eq.  (9).  Right:  Simulations - Errors for  simple and  CB  retrieval. \n\nThe  analysis of simple  retrieval from  the address  i~ yields  with optimal threshold \nsetting 0 = k the add error rate, i.e,  the expectation of spurious ones: \n\n& =  (m - b)Prob [r ~ k]  , \n\n(7) \n\n\f678 \n\nF.  T.  Sommer and G.  Palm \n\nwith  the  binomial  random  variable  Prob[r=l]  =  B(Lit'I,Pt}I,  where  B(n,p),  := \n(7)pl(1  - p)n-l.  a denotes  the  add  error  rate  and  k  = lit'l - a the  number  of \ncorrect  1-s in the address. \n\nFor  the first  step  of CB-retrieval a  lower  bound of the add  error rate a(l)  can  be \nderived by the analysis of CB-retrieval with fixed  address x(O)  =  iIJ.  and the perfect \nlearning  pattern  ylJ.  as  starting patterns in the  y-Iayer.  In this  case the  add  error \nrate is: \n\n(8) \n\nwhere the random variables rl  and r2  have the distributions: \nProb [rl  =  lib] =  B(k, PI),  and Prob [r2  = 1]  =  B(ab, (PI )2) l\"  Thus, \n\na(l) ~ (m - b) L B(k, PdsBS [ab, (PI )2, (k - s)b)  , \n\nk \n\n(9) \n\n8=0 \n\nwhere BS [n,p, t]  := L:~t B(n,p),  is  the binomial sum. \n\nIn  Fig.  1  the  analytic  results  for  the first  step  (7)  and  (9)  can  be  compared  with \nsimulations  (left  versus right  diagram) .  In the experiments simple  retrieval is  per(cid:173)\nformed  with  threshold  ()  =  k.  CB-retrieval  is  iterated  in  the  y-Iayer  (with  fixed \naddress x)  starting with three randomly chosen 1-s from the simple retrieval result \nyt'.  The iteration is stopped, if a  stable pattern at threshold e =  bk  is  reached. \nThe memory capacity can be calculated per pattern component under the assump(cid:173)\ntion  that  in the memory  patterns each  component  is  independent,  i.e.,  the proba(cid:173)\nbilities for  a  1 are p =  a/n or q =  b/m respectively,  and the probabilities of an add \nand  a  miss error are simply the renormalized rates denoted by a', {3'  and a', {3'  for \nx-patterns and  by,', 6'  for  y-patterns.  The information about the stored pattern \ncontained  in  noisy  initial  or  retrieved  patterns is  then  given  by  the  transinforma(cid:173)\ntion  t(p,a',{3')  :=  i(p)  -i(p,a',{3'),  where  i(p)  is  the  Shannon  information,  and \ni (p, a', {3')  the conditional information.  The heteroassociative mapping is evaluated \nby  the  output  capacity:  A(a', {3')  := Mm t(q, ,', 6')/mn  (in  units  bit/synapse).  It \ndepends on the initial noise since the performance drops with growing initial errors \nand  assumes the maximum, if no fault tolerance is  provided, that is,  with noiseless \ninitial  patterns, see  Fig.  2.  Autoassociative completion  of a  distorted x-pattern  is \nevaluated by the completion capacity:  C(a', {3')  := Mn(t(p, a', {3')-t(p, a', {3'))/mn. \nA  BAM  maps  and  completes  at the same time  and  should  be therefore evaluated \nby the search  capacity S  := C  + A. \nThe asymptotic capacity of the Willshaw model is  strikingly high:  The completion \ncapacity (for autoassociation) is C+  =  In[2] /4, the mapping capacity (for heteroas(cid:173)\nsociation  with input  noise)  is  A+  =  In[2] /2 bit/syn [Pal91]'  leading to a  value for \nthe search capacity of (3  In[2])/4 =  0.52 bit/syn.  To estimate S for general retrieval \nprocedures one  can  consider  a  recognition  process  of stored  patterns in  the  whole \nspace  of sparse  initial  patterns;  an  initial  pattern  is  \"recognized\",  if it  is  invari(cid:173)\nant  under  a  bidirectional retrieval  cycle.  The so-called  recognition  capacity of this \nprocess is  an upper bound of the completion capacity  and it had been  determined \nas  In [2J/2,  see  [PS92].  This  is  achieved  again  with  parameters  M, p, q  providing \nA = In[2] /2 yielding  In[2]  bit/syn as upper bound of the asymptotic search capac(cid:173)\nity.  In summary,  we  know  about the asymptotic search capacity of the CB-model: \n0.52  ::;  S+  ::;  0.69 bit/syn.  For experimental results, see  Fig.  4. \n\n\fBidirectional Retrieval from Associative Memory \n\n679 \n\n4  EXPERIMENTAL RESULTS \n\nThe  CB  model  has  been  tested  in  simulations  and  compared  with  the  Willshaw \nmodel (simple retrieval)  for  addresses with random noise  (Fig.  2)  and for  addresses \ncomposed  by two learning patterns  (Fig.  3).  In Fig. 2 the widely enlarged range of \nhigh  qualtity retrieval in  the CB-model is  demonstrated for  different  system sizes. \n\n6 \n5 \n4 \n3 \n2 \n1 \n0 \n\noutput miss errors \n\n10 \n\nsimple r.  .. \". \n\nCB\u00b7r.  -\n\n: \n.. : \" \n\n.' \n\n5  10  15  20  25  30 \n\n8 \n\n6 \n4 \n\n2 \n\n0 \n\n....... \n\n, .. ~ '\" \n.' \n\n, .. \" ... \n\n::' \n\n5  10  15  20  25  30 \n\noutput add errors \n\nsimple r.  \" .\" \nCB\u00b7r.-\n\n6  r--..--.--,--.----.---,  14  rcr-..---,----.~----, \n5 \n4 \n3 \n2 \n\n.:'1 ...... \u00b7 \n\n.,.,.; , \u2022 .-:: .... os\u00b7,\u00b7 \n\n/\" \n\n/\", \n\n12 \n10 \n8 \n6 \n4 \n2 \n\n',' \n\n1 \no ~'--'--'--'--1----'-\"\"'-'-----'  0  L-.C..:....I..!.-...I.-_~=:::::::..J \n5  1015202530 \n\n5  10  15  20  25  30 \n\n101214161820  101214161820 \n\n7 \n6 \n5 \n4 \n3 \n2 \n1 \n0 \n\n10 \n8 \n\n6 \n4 \n2 \n0 \n\n~ \n60 \n\n7 \n6 \n5 \n4 \n3 \n2 \n\n0 \n\n10 \n8 \n6 \n\n4 \n\n~ \n\n60 \n40 \n\ntransinformation in output pattern (bit) \n\n2 \n0  L..:....L-I.. ........ \"--I \n101214161820  101214161820 \n\n1 00  r-r--,-,--,--,1 00  r-r--.-.-T'\"\"\"1 \n\n~ \n\n50  ;c;::r;:::::r:::=r::=7---\"J 1 00 \n45  L \n~ \n35 \n30 \n25 \n~ \n15 \n1 0 \n5 \no '---''---'----1---'----'---' \n5  10  15  20  25  30 \n\ns imple r.  ..\". \n\n\u00b7CB-r.  -\n\n20 \n\n...... . \n\n\" . \n\n5 \n\n10  15  20  25  30 \n\n~ \n20 \no \n101214161820  101214161820 \n\n20 \n0  L-.I.---L.--1-.I..--J \n\nFig.  2:  Retrieval  from  addresses  with  random \nnoise.  The x-axis labeling is  as  in Fig.  1.  Small \nsystem with n  =  100, M  =  35 (left), system size \nas in Fig. 1, two trials (right).  Output activities \nadjusted near Iyl  =  k by threshold setting. \n\nFig.  3:  Retrieval from  addresses \ncomposed  by  two  learning  pat(cid:173)\nterns.  Parameters as in right col(cid:173)\numn of Fig. 2, explanation of left \nand right  column, see text. \n\nIn  Fig.  3 the  address  contains one  learning pattern  and  I-components  of a  second \nlearning  pattern  successively  added  with  increasing  abscissae.  On  the  right  end \nof each diagram both patterns are completely superimposed.  Diagrams in  the left \ncolumn  show  errors  and  transinformation,  if retrieval  results  are  compared  with \nthe  learning pattern which is for  li~ I < 20  dominantly addressed.  Simple  retrieval \nerrors  behave  similiar  as  for  random noise  in  the address  (Fig.  2)  while  the  error \nlevel  of CB-retrieval  raises  faster  if more than  7 adds from  the second  pattern are \npresent.  Diagrams  in  the  right  column  show  the  same  quantities,  if  the  retrieval \nresult is  compared with the closest of the two learning patterns.  It can be observed \ni)  that  a  learning pattern is  retrieved  even  if the address  is  a  complete superposi(cid:173)\ntion  and  ii)  if the  second  pattern is  almost  complete  in  the  address  the  retrieved \npattern corresponds in some cases to the second pattern.  However, in all cases CB(cid:173)\nretrieval  yields  one  of the learning  pattern pairs  and  it  could  be  used  to generate \na  good  address  for  further  retrieval  of the  other  by  deletion  of the  corresponding \nI-components in  the original address. \n\n\f680 \n\n0.48 \n0.46 \n0.44 \n0.42 \n0.4 \n0.38  . \n\noutput c.  ..... \nsearchc.  -\n\n. .... \n\n8  10  12  14  16  18 \n\nF.  T.  Sommer and G.  Palm \n\nFig.  4:  Output  and  search  capacity  of  CB  retrieval  in \nbit/syn with x-axis labeling as  in Fig.  2 for  n = m = 2000, \na = b =  10 M  =  20000.  The difference between both curves \nis  the  contribution  due  to x-pattern  completion,  the com(cid:173)\npletion capacity C.  It is  zero for  Ix(O}1  =  10,  if the initial \npattern is  errorfree. \n\nThe search capacity of the CB model in Fig. 4 is close to the theoretical expectations \nfrom  Sect.  3, increasing with input noise due  to the address completion. \n\n5  SPARSE CODING \n\nTo  apply the proposed N AM  model, for  instance, in information retrieval, a coding \nof the data to be accessed into sparse binary patterns is required.  A useful extraction \nof sparse features should take account of statistical data properties and the way the \nuser  is  acting  on  them.  There  is  evidence  from  cognitive  psychology  that  such  a \ncoding  is  typically  quite  easy  to  find.  The  feature  encoding,  where  a  person  is \nextracting feature sets to characterize complex situations by a few  present features, \nis one of the three basic classes of cognitive processes defined  by Sternberg [Ste77]. \nSimilarities in the data are represented  by feature  patterns having a  large number \nof present features in common, that is  a high overlap:  o(x, x'}  := L:i XiX'i'  For text \nretrieval word fragments used in existing indexing techniques can be directly taken \nas  sparse  binary  features  [Geb87].  For  image  processing  sparse  coding  strategies \n[Zet90],  and  neural  models  for  sparse feature  extraction by  anti-Hebbian  learning \n[F6l90]  have been proposed.  Sparse patterns extracted from different data channels \nin heterogeneous data can simply be concatenated and processed simultaneously in \nN AM.  If parts of the original data should  be held in a  conventional memory,  also \nthese addresses have to be represented  by distributed  and sparse patterns in order \nto exploit  the high performance of the proposed NAM. \n\n6  CONCLUSION \n\nA new bidirectional retrieval method (CB-retrieval) has been presented for the Will(cid:173)\nshaw neural associative  memory model.  Our analysis of the first  CB-retrieval step \nindicates  a  high  potential for  error  reduction  and  increased  input  fault  tolerance. \nThe  asymptotic  capacity for  bidirectional  retrieval in  the  binary  Willshaw  matrix \nhas  been  determined  between  0.52  and  0.69  bit/syn.  In  experiments  CB-retrieval \nshowed  significantly  increased  input  fault  tolerance  with  respect  to  the  standard \nmodel  leading  to  a  practical  information  capacity  in  the  order  of  the  theoretical \nexpectations  (0.5  bit/syn).  Also  the segmentation ability of CB-retrieval with  am(cid:173)\nbiguous  addresses  has  been  shown.  Even  at  high  memory  load  such  input  pat(cid:173)\nterns can be decomposed and corresponding memory entries returned individually. \nThe model improvement does not require sophisticated individual threshold setting \n[GW95],  strategies  proposed  for  BAM  like  more  complex  learning  procedures,  or \n\"dummy augmentation\"  in  the pattern coding  [WCM90,  LCL95]. \n\nThe  demonstrated  performance of the  CB-model  encourages  applications  as  mas(cid:173)\nsively  parallel  search  strategies  in  Information  Retrieval.  The  sparse  coding  re(cid:173)\nquirement  has been  briefly  discussed  regarding technical strategies and  psycholog(cid:173)\nical  plausibility.  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Wang,  J.  B.  Cruz,  and J. H. Mulligan.  Two coding stragegies for bidirec(cid:173)\ntional  associative  memory.  IEEE  Trans.  Neural  Networks,  1(1):81-92,  1990. \n\n[Zet90] \n\nC.  Zetsche.  Sparse coding:  the link  between  low  level  vision  and  associative \nmemory.  In  R.  Eckmiller,  G.  Hartmann,  and  G.  Hauske,  editors,  Parallel \nProcessing  in  Neural  Systems  a.nd  Computers.  Elsevier  Science  Publishers  B. \nV.  (North Holland),  1990. \n\n\f", "award": [], "sourceid": 1377, "authors": [{"given_name": "Friedrich", "family_name": "Sommer", "institution": null}, {"given_name": "G\u00fcnther", "family_name": "Palm", "institution": null}]}