{"title": "The Belief in TAP", "book": "Advances in Neural Information Processing Systems", "page_first": 246, "page_last": 252, "abstract": null, "full_text": "The Belief in  TAP \n\nYoshiyuki Kabashima \n\nDept.  of Compt.  IntI.  &  Syst.  Sci. \n\nTokyo Institute of Technology \n\nDavid Saad \n\nNeural  Computing  Research Group \n\nAston  University \n\nYokohama 226,  Japan \n\nBirmingham B4 7ET,  UK \n\nAbstract \n\nWe  show  the  similarity  between  belief propagation  and  TAP,  for \ndecoding  corrupted  messages  encoded  by  Sourlas's  method.  The \nlatter is  a  special  case of the Gallager  error-correcting code, where \nthe code word comprises products of J{ bits selected randomly from \nthe original message.  We examine the efficacy of solutions obtained \nby the two methods for various values of J{ and show that solutions \nfor  J{  2':  3  may  be  sensitive  to  the  choice  of initial  conditions  in \nthe  case  of unbiased patterns.  Good  approximations  are obtained \ngenerally  for  J{ = 2  and  for  biased  patterns  in  the  case  of  J{ 2':  3, \nespecially when  Nishimori's  temperature is  being used. \n\n1 \n\nIntroduction \n\nBelief networks  [1]  are  diagrammatic  representations  of joint probability  distribu(cid:173)\ntions  over  a  set  of  variables.  This  set  is  usually  represented  by  the  vertices  of \na  graph,  while  arcs  between  vertices  represent  probabilistic  dependencies  between \nvariables .  Belief propagation provides  a  convenient  mathematical tool for  calculat(cid:173)\ning iteratively joint probability distributions  between variables  and  have been used \nin a variety of cases,  most recently in the field  of error correcting codes, for  decoding \ncorrupted messages [2]  (for a review of graphical models and their use in the context \nof error-correcting  codes  see  [3]). \n\nError-correcting codes provide a mechanism for retrieving the original message after \ncorruption due to noise  during transmission.  Of a  particular interest to the current \npaper is  an error-correcting code presented by Sourlas  [4]  which is  a  special  case of \nthe Gallager codes  [5].  The latter have been recently re-discovered by MacKay and \nNeal  [2]  and seem to have  a  significant practical potential. \n\nIn  this paper  we  will  examine  the similarities between the belief propagation  (BP) \nand  TAP  approaches,  used  to  decode  corrupted  messaged  encoded  by  Sourlas's \nmethod , and compare the solutions obtained by both approaches to the exact results \nobtained using the replica method [8].  The statistical mechanics approach will  then \n\n\fThe Belie/in TAP \n\n247 \n\nallow  us  to draw some  conclusion  on  the  efficacy  of the  TAP /BP  approach  in  the \ncontext of error correcting codes. \n\nThe paper is  arranged in  the following  manner:  In section  2  we  will  introduce the \nencoding method and describe the decoding task.  The Belief Propagation approach \nto the decoding process will be introduced in section 3 and  will  be compared to the \nTAP approach for  diluted spin systems in section 4.  Numerical solutions for  various \ncases  will  be presented  in  section  5 and  we  will  summarize our results  and  discuss \ntheir implications  in  section  6. \n\n2  The decoding  problem \nIn  a  general  scenario,  a  message  represented  ~ an  N  dimensional  binary  vector e \nis  encoded  by a  vector  JO  which is  then transmitted through  a noisy  channel with \nsome  flipping  probability  p  per  bit.  The  received  message  J  is  then  decoded  to \nretrieve the original  message.  Sourlas's  code  [4],  is  based  on  encoded message  bits \n. . iK = ~il ei 2  . . .  eiK , taking the product of different  J{  message sites \nof the form  JPl,i 2 \nfor  each code  word  bit . \n\nIn  the statistical mechanics  approach  we  will  attempt to retrieve the original  mes(cid:173)\nsage  by exploring the ground state of the following  Hamiltonian which corresponds \nto the preferred state of the system in  terms of 'energy' \n\n1{=- L  Ah , ... iK)  J(il , .. iK)  Si l \u00b7 \u00b7 .SiK  - F/f3LSk  , \n\n(1) \n\n(i1, ... iK) \n\nk \n\nwhere  S is  an  N  dimensional binary vector of dynamical variables and A  is  a sparse \ntensor with C  unit elements  per index (other elements  are zero),  which determines \nthe  components  of JO.  The last  term on  the right is  required  in  the  case of sparse \n(biased)  messages  and  will  require  assigning  a  certain  value  to  the  additive  field \nF / f3,  related  to the prior belief in the Bayesian framework. \nThe  statistical  mechanical  analysis  can  be  easily  linked  to  the  Bayesian  frame(cid:173)\nwork  [4]  in  which  one  focuses  on  the  posterior  probability  using  Bayes  theorem \nP(SIJ)\",,-, IT!' P(J!'IS) Po(S) where jJ runs over the message components and Po(S) \nrepresents  the prior .  Knowing the posterior one can  calculate the typical retrieved \nmessage elements  and  their alignment,  which correspond to the  Bayes-optimal de(cid:173)\ncoding.  The logarithms of the likelihood  and  prior terms are directly related to the \nfirst  and second  components of the  Hamiltonian  (Eq.l). \n\nOne should  also note that A(il , . iK) J(il , . i K) represents  a  similar  encoding scheme \nto that of Ref.  [2]  where a sparse matrix with J{ non-zero elements per row multiplies \nthe original  message e and the resulting vector, modulo 2,  is  transmitted. \nSourlas  analyzed  this  code  in  the  cases  of  J{  =  2  and  J{  -+  00,  where  the  ratio \nC / J{  -+ 00 ,  by  mapping  them  onto  the  SK  [9]  and  Random  Energy  [10]  models \nrespectively.  However, the ratio R = J{ / C  constitutes the code rate and the scenarios \nexamined  by Sourlas  therefore  correspond  to  the  limited  case  of a  vanishing  code \nrate.  The  case  of finite  code  rate,  which  we  will  consider  here ,  has  only  recently \nbeen  analyzed  [8]. \n\n3  Decoding by belief propagation \n\nAs  our goal, of calculating the posterior of the system P( S IJ) is  rather difficult , we \nresort to the methods of BP, focusing on the calculation of conditional probabilities \nwhen  some elements of the system are set  to specific  values  or removed. \n\n\f248 \n\nY  Kabashima and D.  Saad \n\nThe approach adopted in  this case,  which  is  quite similar to the practical approach \nemployed in the case of Gallager codes [2],  assumes a two layer system corresponding \nto the elements of the  corrupted  message  J  and the dynamical variables  5  respec(cid:173)\ntively,  defining  conditional  probabilities  which relate elements  in  the two layers: \n\nr~1 \n\n(2) \nP(JI'ISI=X,{JII~I'}) =  L  P(JI'ISI=X,{Sk#})  P({Sk~dl{JII~I'})  , \n\n{ Sk;tz} \n\nwhere  the  index  J.l  represents  an  element  of the  received  vector  message  J,  con(cid:173)\nstituted  by  a  particular  choice  of indices  i 1 , .. . iK,  which  is  connected  to  the  cor(cid:173)\nresponding  index  of  5  (l  in  the  first  equation),  i.e.,  for  which  the  corresponding \nelement A(i1, ... iK)  is  non-zero;  the notation  {Sk~d refers  to  all  elements of 5,  ex(cid:173)\ncluding  the  I-th  element,  which  are  connected  to the  corresponding  index of J \n(J.l \nin  this  case for  the  second  equation);  the index  x  can  take values of \u00b11.  The con(cid:173)\nditional  probabilities  q~1  and  r~1  will  enable  us,  through  recursive  calculations  to \nobtain  an  approximated expression to the posterior. \nEmploying  Bayes  rule  and  the  assumption  that  the  dependency  of  SI  on \nan  element  JII \nis  factorizable  and  vice  versa:  P(SI 1 ,SI 2 \",SIKI{Jvtl'})  = \nnf=lP(Slkl{JIlt/'})  and  P({J/J~I'}  ISI=x)  =  nllt/'P(JIIISI=X,{J(7~II})' \none  can rewrite  a set of coupled equations for  q!/' q;/  , r!1  and  r;/ of the form \n\nq~1 = al'l  PI IT r:1  and  r~1 = L  P (J I' lSI = x, {Sk#})  IT q!Z  ' \n\n(3) \n\n/J~I' \n\n{Sk;tz} \n\nk~1 \n\nwhere  al'l  is  a  normalizing factor  such  that  q~1 + q;/ = 1 and pf  = P (SI = x)  are \nour prior  beliefs  in the value of the source bits  SI. \n\nThis set of equations can be solved iteratively [2]  by updating a coupled set of differ(cid:173)\nence equations for  8ql'I = q~/-q;/ and 8rl'I = r~l-r;/, derived for  this specific  model, \nmaking use of the fact that the variables  r~/' and sub-sequentially the variables  q~/' \ncan be calculated  by exploiting  the relation  r;/ = (1\u00b18rl'l)/2 and  Eq.(3).  At each \niteration  we  can  also  calculate the  pseudo-posterior  probabilities  qf = alPI nil r~/' \nwhere  al  are  normalizing factors,  to determine  the current estimated value of SI. \n\nTwo points that are  worthwhile noting:  Firstly,  the iterative solution  makes  use  of \nthe normalization r~/+r;/ = 1,  which is  not derived from the basic  probability rules \nand  makes implicit assumptions  about the probabilities of obtaining SI = \u00b11 for  all \nelements I.  Secondly,  the  iterative solution  would have  provided the  true posterior \nprobabilities qf  if the graph connecting the message J  and the encoded bits 5  would \nhave been  free  of cycles,  i.e.,  if the graph would  have been a  tree with no  recurrent \ndependencies  among the variables.  The fact that  the framework  provides  adequate \npractical solutions  has only recently been explained  [13]. \n\n4  Decoding by TAP \n\nWe  will  now show  that for  this  particular problem it is  possible  to obtain  a similar \nset of equations from  the  corresponding statistical  mechanics  framework  based  on \nBethe  approximation  [11]  or  the  TAP  (Thouless-Anderson-Palmer)  approach  [12] \nto diluted systems  1  .  In the statistical mechanics approach we  assign  a  Boltzmann \n\n1  The  terminology  in  the  case  of diluted  systems  is  slightly  vague.  Unlike  in  the  case \nof fully  connected systems,  self consistent equations of diluted  systems  cannot  be  derived \n\n\fThe  Beliefin TAP \n\n249 \n\nweight to each set  comprising an encoded message bit J II.  and a dynamical  vector S \n\nWE  (JII.IS)  =  e-{3  9(1I'IS)  , \n\n(4) \nsuch  that  the  first  term  of the  system's  Hamiltonian  (Eq.1)  can  be  rewritten  as \nL II.  g ( Jil.l S) , where the index J..l  runs over all non-zero sites in the multidimensional \ntensor  A.  We  will  now  employ  two  straightforward  assumptions  to  write  a  set  of \ncoupled  equations for  the mean field  q~1 ==  P(511 {Jvtll.})'  which may be identified \nas  the  same variable  as  in  the  belief network  framework  (Eq.2) ,  and  the effective \nBoltzmann weight  weff (J 11.151, {J vtll.}): \n1)  we  assume  a  mean field  behavior for  the  dependence of the dynamical  variables \nS  on a certain realization of the message sites J, i.e.,  the dependence is  factorizable \nand  may be replaced by a  product of mean fields . \n2)  Boltzmann weights  (effective)  for  site 51  are factorizable  with  respect  to  J 11.. \nThe resulting set of equations  are of the form \n\nweff(J1l.  151, {Jvtll.})  =  Tr{Sk;l!z}  WE  (JII.  1 S)  II q~r \n\nqSl \n11.1 \n\n==  CLII.I  pfl  II Weff(Jv  151, {J\"tv})  , \n\nk;tl \n\nvtll. \n\n(5) \n\nwhere  CLII.I  is  a  normalization factor  and pfl  is  our  prior  knowledge  of the  source's \nbias.  Replacing  the  effective  Boltzmann  weight  by  a  normalized  field,  which  may \nbe identified  as  the variable r~1  of Eq.(2),  we  obtain \n\nr~l =  P (51  1 JII.' {Jvtll.}) =  ali.I  weff(J1l.  151, {Jvtll.})  , \n\n(6) \ni.e.,  a set of equations equivalent to Eq.(3).  The explicit expressions of the normal(cid:173)\nization coefficients,  ali.I  and  CLII.I'  are \n\na~l = Tr{s}  WE  (JII.IS) II q~f \n\nk;tl \n\nand \n\n(7) \n\nThe  somewhat  arbitrary  use  of the  differences  oqll.l  = (5i}q  and  Dril.l  = (5i}r  in \nthe  BP  approach  becomes  clear  form  the  statistical  mechanics  description,  where \nthey represent the expectation values of the dynamical variables with respect to the \nfields .  The statistical  mechanics  formulation  also  provides  a  partial  answer  to the \nsuccessful  use of the  BP  methods  to loopy systems , as  we  consider  a finite  number \nof steps on  an  infinite  lattice  [14].  However, it  does  not  provide  an explanation  in \nthe case of small systems  which should be examined  using other methods . \n\nThe formulation  so far  has  been general;  however,  in  the case  of Sourlas 's code  we \ncan make use of the explicit expression for  g to derive the relat\\on between q~l, r;l , \noqll.l  and  Dril.l  as  well  as  an  explicit expression for  WE  (JII.IS,,8) \n\n,  r~l = ~ (1 + or1l.151)  and \nq~l  ==  ~ (1 + oq1l.151) \nWE  (JII.IS ,,8)  = ~ cosh ,8JII.  (1 + tanh,8J II.  II 51) \n\nI H(II.) \n\n, \n\n(8) \n\n(9) \n\nby  the perturbation expansion  of the mean field  equations  with  respect  to  Onsager  reac(cid:173)\ntion  fields  since these  fields  are  too large in  diluted  systems.  Consequently,  the  resulting \nequations  are  different  than  those  obtained  for  fully  connected  systems  [12].  We  termed \nour  approach TAP, following  the convention for  the Bethe approximation when applied to \ndisordered  systems  subject  to mean  field  type random interactions. \n\n\f250 \n\nY.  Kabashima  and D.  Saad \n\nwhere  .C(J.l)  is  the  set  of all  sites  of S  connected  to  J/.I '  i.e. ,  for  which  the  corre(cid:173)\nsponding  element  of the  tensor  A  is  non-zero.  The  explicit  form  of the  equations \nfor  8q/.ll  and 8r/.ll  becomes \n\n8rjjl=tanhf3J/.I  II 8q/.ll \n\nIEC,(/.I)/I \n\nand  8q/.ll =tanh (  L  tanh- 1 8rv i  + F)  , (10) \n\nvEM(l)//.I \n\nwhere  M(l)/ J.l  is  the  set  of  all  indices  of the  tensor  J ,  excluding  J.l,  which  are \nconnected to the vector site I;  the external field  F  which previously appeared in  the \nlast term of Eq. (1)  is  directly related  to our  prior belief of the message  bias \n\n1 \n\npfl  =  \"2  (1  +  tanh FSI)  . \n\n(11) \n\nWe  therefore  showed  that there  is  a  direct  relation  between the  equations  derived \nfrom  the BP  approach and from TAP in  this particular case.  One should  note that \nthe TAP  approach  allows  for  the  use  of finite  inverse-temperatures  f3  which  is  not \nnaturally  included  in  the BP  approach. \n\n5  Numerical  solutions \n\nTo  examine  the  efficacy  of TAP /BP  decoding  we  used  the  method  for  decoding \ncorrupted messages encoded by the Sourlas scheme [4],  for which we have previously \nobtained  analytical  solutions  using  the  replica  method  [8].  We  solved  iteratively \nEq.(10)  for  specific  cases  by  making  use  of differences  8qJjI  and  8r/.ll  to  obtain  the \nvalues of q~l and r'N  and of the magnetization  M. \nNumerical solutions  of 10  individual  runs for  each  value  of the flip  rate  p  starting \nfrom  different  initial  conditions,  obtained  for  the  case  f{ =  2  and  C  =  4,  different \nbiases  (J = pi =  0.1, 0.5 - the  probability of +1  bit in  the  original  message e)  and \ntemperatures  (T =  0.26, Tn)  are  shown  in  Fig .  1a.  For  each  run ,  20000  bit  code \nwords  JO  were  generated  from  10000  bit  message e using  a  fixed  random  sparse \ntensor  A .  The  noise  corrupted  code  word  J  was  decoded  to retrieve  the  original \nmessage e.  Initial conditions are set to 8r /.II  = 0 and 8q/.ll = tanh F reflecting the prior \nbelief; whenever the TAP /BP approach was successful in predicting the theoretical \nvalues  we  observed  convergence  in  most  runs  corresponding  to  the  ferromagnetic \nphase  while  almost  all  runs  at  low  temperatures  did  not  converged  to  a  stable \nsolution above the critical flip-rate  (although the magnetization  M  did  converge as \none  may expect) .  We  obtain good  agreement  between the TAP /BP solutions  and \nthe  theoretical  values  calculated  using  the  methods  of  [8]  (diamond  symbols  and \ndashed  line  respectively) .  The  results  for  biased  patterns  at  T =  0.26  presented  in \nthe form of mean values  and standard deviation , show a sub-optimal improvement \nin  performance  as  expected.  Obtaining solutions  under  similar  conditions  but  at \nNishimori's  temperature - l/Tn  = 1/2In[(1 - p)/p]  [7],  we see  that pattern sparsity \nis  exploited  optimally  resulting  in  a  magnetization  M  ~ 0.8  for  high  corruption \nrates , as  Tn  simulates accurately the loss  of information  due to channel noise  [6 , 7]; \nresults  for  unbiased  patterns  (not shown)  are  not  affected  significantly by the use \nof Nishimori's temperature. \n\nThe replica-based theoretical solutions [8]  indicate a profoundly different behaviour \nfor  f{ =  2  in  comparison  to  other  f{  values.  We  therefore  obtained  solutions  for \nJ{  =  5  under  similar  conditions  (which  are  representative  of results  obtained  in \nother  cases  of  f{ # 2).  The  results  presented  in  Fig.  1b ,  in  terms  of means  and \nstandard  deviation  of 10  individual  runs  per flip  rate value p, are  less  encouraging \nas the iterative solutions  are sensitive to the choice of initial conditions and  tend to \n\n\fThe Beliefin TAP \n\n251 \n\n1.2  r---,-----.-----.---,-------, \n\n1.2  r---...,------,------.----,----, \n\na) K=2 \n\n.~~-\nO~i\"\\\\ '\"'--\"'II8fiIllmllIat!J8!jlilalll!. \n0\" \nI \n.. \n\u00b0 s~  \\ \n\nI\\ \n1 \ni) \n\n) \nv'l, \n)o~ \nT =0.26, Unbiased  08: \n~=O.5~  9 \n' l .   T=O.26, Biased \n~ \n(I \n=O.l~ \no~ \no \nP \n\u00b0SoP \n\nt~ \n'1.T \n\u00b7!fi \n\n~'f.t' \n\ni \nI \nMn, Biased \n(1=O.l~ \n\n1 \n\n, \n\nO.S \n\nO 6 \n\u2022 \n\nO 4 \n. \n\n0.2 \n\n.: \n\no  8o:h!l. \n\n?~~ \n\nf~ \n\no \no \n\n0.1 \n\n0.2 \n\np \n\n0.3 \n\n0.4 \n\n0.5 \n\nb) K=5 \n\n... ~ \n\nMn, Biased \n(1=O.l~ \n\ni  ,-,~1~ \n/  i  1.001 \n\nT:Tn, Unbiased i \n~=O.5~  , \n\n~~1K-iiooh \n\n0.999 \n\n0.998 \n\n0.997 \n\n0.996 \n\nO.S \n\n0.6 \n\n0.4 \n\n0.2 \n\n0.995  '---'-~~~~---LJ \no 0.020.040.060.06  0.1 0.120.14 \no ~'MttItMi.u., ......... tttftl\"\"Itttt\"'\" \no \n0.4 \n\n0.2 \n\n0.3 \n\n0.1 \n\n0.5 \n\np \n\n(a)  For  K  = 2, \nFigure  1:  Numerical  solutions  for  M  and  different  flip  rate  p. \ndifferent  biases  (f = pi = 0.1, 0.5)  and  temperatures  (T = 0.26, Tn).  Results for  the \nunbiased patterns  are shown  as  raw data (10  runs per flip  rate value p  - diamond), \nwhile  the  theoretical  solution  is  marked  by  the  dashed  line.  Results  for  biased \npatterns are presented by their mean and standard deviation, showing a suboptimal \nperformance as expected for T= 0.26 and an optimal one at Nishimori's temperature \n-Tn.  The  standard  deviation  is  significantly smaller  than  the symbol  size.  Figure \n(b)  shows  results  for  the  case  K  = 5  and  T  = Tn  in  similar  conditions  to  (a). \nAlso  here  iterative  solutions  may  generally  drift  away  from  the theoretical  values \nwhere  temperatures  other  than  Tn  are  employed  (not  shown);  using  Nishimori's \ntemperature  alleviates  the  problem  only  in  the  case  of biased  messages  and  the \nresults  are in  close  agreement with the theoretical solutions (inset - focusing on low \np  values). \n\nconverge  to sub-optimal  values  unless  high  sparsity  and  the  appropriate  choice  of \ntemperature  (Tn)  forces  them to the  correct  values,  showing  then good  agreement \nwith the theoretical results (solid line, see inset).  This phenomena is indicative of the \nfact  that the  ground  state of the non-biased  system is  macroscopically  degenerate \nwith multiple equally good ground states. \n\nWe  conclude that the TAP /BP approach may be highly useful in the case of biased \npatterns but may lead to errors for  unbiased  patterns and  K 2: 3,  and  that the  use \nof the appropriate temperature,  i.e.,  Nishimori's temperature, enables one to obtain \nimproved  results,  in  agreement with results  presented elsewhere  [4,  6,  7]. \n\n\f252 \n\nY.  Kabashima and D.  Saad \n\n6  Summary and discussion \n\nWe compared the use of BP to that of TAP for decoding corrupted messages encoded \nby Sourlas's method to discover that in this particular case the two methods provide \na similar set of equations.  We then solved the equations iteratively for specific cases \nand  compared  the  results  to  those obtained  by  the replica method.  The solutions \nindicate  that  the  method  is  particularly  useful  in  the  case  of biased  messages  and \nthat  using  Nishimori's  temperature  is  highly  beneficial;  solutions  obtained  using \nother temperature values  may be  sub-optimal.  For non-sparse  messages  and  l{ 2: 3 \nwe  may obtain erroneous solutions  using  these methods. \n\nIt would be desirable to explore whether the similarity in the equations derived using \nTAP and  BP is  restricted  to this particular case or whether there is  a  more general \nlink  between  the  two  methods.  Another  important  question  that  remains  open \nis  the  generality  of our  conclusions  on  the  efficacy  of these  methods  for  decoding \ncorrupted messages,  as  they are currently being applied in  a  variety of state-of-the(cid:173)\nart coding schemes (e.g., [2,3]).  Understanding the limitations ofthese methods and \nthe  proper way to use  them in  general,  especially  in the  context of error-correcting \ncodes,  may be highly  beneficial to  practitioners. \n\nAcknowledgment This work was partially supported by the RFTF program of the  JSPS \n(YK)  and  by  EPSRC grant GR/L19232  (DS). \n\nReferences \n\n[1]  J.  Pearl,  Probabilistic  Reasoning in Intelligent  Systems:  Networks  of Plausible \n\nInference  (Morgan  Kaufmann)  1988. \n\n[2]  D.J .C. MacKay and  R.M.  Neal,  Elect.  Lett.,  33, 457  and preprint (1997). \n[3]  B.J. Frey,  Graphical Models  for Machine  Learning  and Digital Communication \n\n(MIT Press),  1998. \n\n[4]  N.  Sourlas,  Nature,  339, 693  (1989)  and  Europhys.  Lett.,  25,  159  (1994). \n[5]  R.G.  Gallager,  IRE  Trans.  Info.  Theory,  IT-8, 21  (1962). \n[6]  P.  Rujan,  Phys.  Rev.  Lett., 10, 2968  (1993) . \n[7]  H.  Nishimori,J.  Phys.  C,  13,4071  (1980)  and  J.  Phys.  Soc.  of Japan,  62,  1169 \n\n(1993). \n\n[8]  Y.  Kabashima and  D.  Saad,  Europhys.  Lett., 45, in  press  (1999). \n[9]  D.  Sherrington and  S.  Kirkpatrick,  Phys.  Rev.  Lett., 35,  1792  (1975). \n[10]  B.  Derrida,  Phys.  Rev.  B,  24,  2613  (1981). \n[11]  H.  Bethe,  Proc.  R.  Soc.  A, 151,  540  (1935) . \n[12]  D.  Thouless,  P.W.  Anderson  and  R.G.  Palmer,  Phil.  Mag.,  35, 593  (1977). \n[13]  Y.  Weiss,  MIT preprint CBCL155  (1997). \n[14]  D.  Sherrington and  K.Y.M . Wong  J.  Phys.  A, 20,  L785  (1987). \n\n\f", "award": [], "sourceid": 1570, "authors": [{"given_name": "Yoshiyuki", "family_name": "Kabashima", "institution": null}, {"given_name": "David", "family_name": "Saad", "institution": null}]}