{"title": "Clustering data through an analogy to the Potts model", "book": "Advances in Neural Information Processing Systems", "page_first": 416, "page_last": 422, "abstract": null, "full_text": "Clustering data through an analogy  to \n\nthe Potts model \n\nMarcelo Blatt,  Shai Wiseman and Eytan  Domany \n\nDepartment of Physics  of Complex Systems, \n\nThe Weizmann Institute of Science,  Rehovot  76100,  Israel \n\nAbstract \n\nA  new  approach for  clustering  is  proposed.  This  method is  based \non an analogy to a  physical model;  the  ferromagnetic  Potts model \nat thermal equilibrium is  used  as an analog computer for  this hard \noptimization problem .  We  do not  assume  any structure  of the  un(cid:173)\nderlying distribution of the data.  Phase space of the Potts model is \ndivided  into  three  regions; ferromagnetic,  super-paramagnetic and \nparamagnetic phases.  The  region  of interest  is  that  corresponding \nto the super-paramagnetic one, where  domains of aligned spins ap(cid:173)\npear.  The range of temperatures  where  these structures  are  stable \nis  indicated  by  a  non-vanishing magnetic susceptibility.  We  use  a \nvery  efficient  Monte  Carlo  algorithm  to  measure  the  susceptibil(cid:173)\nity  and  the  spin  spin  correlation  function.  The  values  of the  spin \nspin  correlation  function,  at  the  super-paramagnetic  phase,  serve \nto identify the  partition of the  data points into clusters. \n\nMany natural phenomena can be viewed as optimization processes,  and the drive to \nunderstand and analyze them yielded powerful  mathematical methods.  Thus when \nwishing to solve a hard optimization problem, it may be advantageous to apply these \nmethods  through  a  physical  analogy.  Indeed,  recently  techniques  from  statistical \nphysics  have  been  adapted for  solving hard optimization problems  (see  e.g.  Yuille \nand Kosowsky,  1994).  In  this work  we formulate the problem of clustering in  terms \nof a  ferromagnetic  Potts spin  model.  Using  the  Monte  Carlo method  we  estimate \nphysical quantities such  as the spin spin correlation function  and the susceptibility, \nand  deduce from them the  number of clusters  and cluster  sizes. \nCluster  analysis  is  an  important technique  in exploratory  data analysis  and  is  ap(cid:173)\nplied in a variety of engineering and scientific disciplines.  The problem of partitionaZ \nclustering can  be formally stated  as  follows.  With everyone of i  =  1,2, ... N  pat(cid:173)\nterns  represented  as  a  point  Xi  in  a  d-dimensional  metric  space,  determine  the \npartition  of these  N  points  into  M  groups,  called  clusters,  such  that  points  in  a \ncluster are more similar to each other than to points in different  clusters.  The value \nof M  also has  to be  determined. \n\n\fClustering Data  through  an  Analogy  to  the Potts  Model \n\n417 \n\nThe  two  main approaches  to partitional clustering  are  called  parametric and  non(cid:173)\nparametric.  In  parametric approaches  some knowledge of the  clusters'  structure  is \nassumed  (e.g .  each  cluster  can  be  represented  by  a  center  and  a  spread  around \nit) .  This assumption is  incorporated in a  global  criterion.  The goal is  to assign  the \ndata points so  that  the  criterion  is  minimized .  A  typical example is  variance  min(cid:173)\nimization (Rose, Gurewitz,  and  Fox,  1993) .  On  the other hand,  in  non-parametric \napproaches  a  local  criterion is  used  to  build  clusters  by  utilizing local  structure  of \nthe  data.  For  example,  clusters  can  be formed  by  identifying high-density  regions \nin  the  data space  or  by  assigning a  point  and its  K -nearest  neighbors  to the same \ncluster.  In  recent  years  many  parametric  partitional  clustering  algorithms rooted \nin statistical physics  were  presented  (see  e.g.  Buhmann and  Kiihnel  , 1993).  In the \npresent  work  we  use  methods of statistical physics  in non-parametric clustering. \n\nOur  aim is  to  use  a  physical  problem as  an analog to  the  clustering  problem.  The \nnotion of clusters  comes very  naturally in  Potts spin models  (Wang and Swendsen, \n1990) where clusters are closely related to ordered regions of spins.  We place a Potts \nspin variable Si  at each point Xi  (that represents one of the patterns), and introduce \na  short  range  ferromagnetic  interaction  Jij  between  pairs of spins,  whose  strength \ndecreases  as  the inter-spin distance Ilxi - Xj\"  increases.  The system is  governed by \nthe  Hamiltonian (energy function) \n\n1i =  - L  hj D8,,8j \n\n<i,j> \n\nSi  = 1 . .. q , \n\n(1) \n\nwhere  the notation < i, j  >  stands for  neighboring points  i  and j  in  a sense  that is \ndefined  later.  Then  we  study  the ordering  properties  of this  inhomogeneous Potts \nmodel. \n\nAs  a  concrete  example, place a  Potts spin  at each  of the  data points of fig.  1. \n\n~~--~------~--------~--------~------~--------~------~--~ \n\n\u00b730 \n\n\u00b720 \n\n-10 \n\n10 \n\n20 \n\n30 \n\nFigure  1:  This  data set  is  made of three  rectangles,  each  consisting  of 800  points \nuniformly distributed , and a  uniform rectangular  background of lower  density,  also \nconsisting  of  800  points.  Points  classified  (with  Tclus  =  0.08  and  ()  =  0.5)  as \nbelonging to the  three  largest  clusters  are  marked by  crosses,  squares and x's.  The \nfourth cluster is of size 2 and all others are single point clusters marked by triangles . \n\nAt  high  temperatures  the  system  is  in  a  disordered  (paramagnetic)  phase.  As \nthe  temperature  is  lowered,  larger and  larger  regions  of high  density  of points  (or \nspins)  exhibit local  ordering,  until a  phase  transition occurs  and spins in  the  three \nrectangular high density regions become completely aligned (i. e.  within each region \nall  Si  take  the same value - super-paramagnetic phase) . \nThe aligned regions define the clusters which we wish to identify.  As the temperature \n\n\f418 \n\nM.  BLATT, S.  WISEMAN, E.  DOMANY \n\nis  further  lowered,  a  pseudo-transition  occurs  and  the system  becomes  completely \nordered  (ferromagnetic). \n\n1  A  mean field  model \n\nTo support  our main idea,  we  analyze  an idealized  set  of points where  the division \ninto natural classes is  distinct.  The points are divided into M  groups.  The distance \nbetween any two points within the same group is  d1  while the distance between any \ntwo points belonging to different groups is d2  > d1  (d can be regarded as a similarity \nindex).  Following our main idea,  we  associate  a  Potts spin  with each  point and  an \ninteraction  J1  between  points  separated  by  distance  d1  and  an  h  between  points \nseparated  by  d2 ,  where  a ~ J2  < J 1 \u2022  Hence  the  Hamiltonian (1)  becomes; \n\n1{  =  - ~ L L 6~; ,~j  - ~ L L 6s; ,sj \n\n/10 \n\ni<j \n\n/1o<V \n\ni ,j \n\nsi  =  1, ... , q  , \n\n(2) \n\nwhere  si  denotes  the  ith  spin  (i =  1, ... , ~) of the  lJth  group  (lJ  =  1, ... , M). \nFrom standard  mean field  theory  for  the  Potts  model  (Wu ,  1982)  it  is  possible  to \nshow  that  the  transition from  the  ferromagnetic  phase  to  the  paramagnetic  phase \nis  at  Tc  = 2M (qJ.)~Og(q-l)  [J1 + (M - 1)h]  . The  average  spin  spin  correlation \nfunction,  6~,,~ j  at the paramagnetic phase is t for  all points Xi  and Xj;  i. e.  the spin \n\nvalue at each point is  independent of the others.  The ferromagnetic phase is further \ndivided into two  regions.  At  low  temperatures,  with high  probability, all spins  are \naligned;  that is  6~.,sJ  ~ 1 for  all i  and j.  At intermediate temperatures,  between T* \nand Tc,  only spins of the same group lJ  are aligned with high probability; 6~\" ~'-:  ~ 1, \nwhile spins  belonging to different  groups,  Jl  and  lJ,  are independent;  6~1\"  s~ ~ 1 . \nq \nThe  spin  spin  correlation  function  at  the  super-paramagnetic  phase  can  be  used \nto  decide  whether  or  not  two  spins  belong  to  the  same  cluster.  In  contrast  with \nthe  mere  inter-point  distance,  the  spin spin  correlation function  is  sensitive  to  the \ncollective  behavior  of the  system  and  is  therefore  a  suitable  quantity  for  defining \ncollective structures  (clusters). \nThe transition temperature T* may be calculated and shown  to be  proportional to \nJ2 ;  T*  =  a(N, M, q)  h. In  figure  2  we  present  the  phase  diagram,  in  the  (~, ~) \nplane, for  the  case  M  = 4,  N  = 1000 and  q = 6. \n\n\u2022  ' 1 \n\n. '   J \n\n1e-01  ~ _____  --,-____  ~~ \"1 \n\nparamagnetic \n\n/ '  \n\n/ \nsuper-paramagnet~s-,'-' \n\n1e\"()2 \n\n.., \nf:: \n\n/ \n\"\",,; \n\n.(cid:173)\n-' \n\n;\" .. , \n\n1e-03 \n\n./,' \n\n, ...... ', .. ' \n\nferromagnetic \n\n1e..()4~--~~~~--~~----~~~ \n1e+OO \n\n1e\"()4 \n\nle-02 \n\n1e-03 \n\n1e..()S \n\nle..()1 \n\nJ2JJl \n\nFigure  2:  Phase  diagram \nof  the  mean  field  Potts \nmodel  (2)  for \nthe  case \nM  = 4,  N  =  1000 and q = \n6.  The  critical  tempera(cid:173)\nture Tc  is indicated  by the \nsolid  line,  and  the  transi(cid:173)\ntion  temperature  T*,  by \nthe  dashed  line. \n\nThe  phase  diagram fig.  2 shows  that the  existence  of natural  classes  can  manifest \nitself  in  the  thermodynamic  properties  of  the  proposed  Potts  model.  Thus  our \napproach  is  supported,  provided  that  a  correct  choice  of the  interaction strengths \nis  made. \n\n\fClustering Data through  an  Analogy  to the  Potts  Model \n\n419 \n\n2  Definition  of local  interaction \n\nIn order to minimize the intra-cluster interaction it is  convenient to allow an interac(cid:173)\ntion only bet.ween  \"neighbors\".  In  common \\ .... ith  other  \"local  met.hods\" , we  assume \nthat there is  a  'local length scale'  '\" a,  which  is  defined  by  the  high  density  regions \nand is  smaller  than  the  typical  distance  between  points  in  the  low  density  regions. \nThis  property  can  be  expressed  in  the  ordering  properties  of the  Potts system  by \nchoosing a  short range interaction .  Therefore  we  consider  that each  point interacts \nonly with  its neighbors  with interaction strength \n\n__  1 \n\nJ ji  - R exp \n\nJij  -\n\n(!lXi-Xj!l2) \n-\n\n2a 2 \n\n. \n\n(3) \n\nTwo points,  Xi  and Xj,  are defined as neighbors if they have a mutual neighborhood \nvalue  J{;  that is,  if Xi  is  one of the  J{  nearest  neighbors  of Xj  and  vice-versa.  This \ndefinition  ensures  that  hj  is  symmetric;  the  number  of  bonds  of any  site  is  less \nthan  J{.  We  chose  the  \"local  length  scale\",  a,  to  be  the  average  of all  distances \nIlxi - Xj II  between  pairs  i  and  j  with  a  mutual  neighborhood  value  J{.  R is  the \naverage  number  of neighbors  per  site;  i. e it  is  twice  the  number  of non  vanishing \ninteractions,  Jij  divided  by  the  number of points N  (This  careful  normalization of \nthe  interaction  strength  enables  us  to estimate the critical  temperature Tc  for  any \ndata sample). \n\n3  Calculation  of thermodynanlic quantities \n\nThe  ordering  properties  of  the  system  are  reflected  by  the  susceptibility  and  the \nSpill  spin  correlation  functioll  D'<\"'<J'  where  -.. -. stands for  a  thermal  average.  These \nquantities  can  be  estimated  by  averaging  over  the  configurations  genel'ated  by  a \nMonte  Carlo  procedure.  We  use  the  Swendsen-Wang  (Wang  and  Swendsen,  1990) \nMonte Carlo algorithm for  the Potts model (1) not only because of its high efficiency, \nbut also  because  it  utilizes  the  SW  clusters.  As  will  be explained  the  SW  clusters \nare  strongly  connected  to the clusters  we  wish  to identify.  A  layman's explanation \nof the  method  is  as  follows.  The  SW  procedure  stochastically  identifies  clusters \nof  aligned  spins,  and  then  flips  whole  clusters  simultaneously.  Starting  from  a \ngiven  spin  configuration,  SW  go  over  all  the  bonds  between  neighboring  points, \nand either  \"freeze\"  or  delete  them.  A  bond  connecting  two neighboring sites  i  and \n\nj,  is  deleted  with  probability  p~,j  =  exp( -* 63  .. 3 J and  frozen  with  probability \n\np?  = 1 - p~,j.  Having  gone  over  all  the  bonds ,  all  spins  which  have  a  path  of \nfrozen  bonds  connecting them are identified as  being in  the same SW  cluster.  Note \nt.hat,  according  to the  definition of p~,j, only spins of the same value can be frozen \nin the same SW cluster.  Now  a  new  spin configuration  is  generated by  drawing, for \neach  cluster,  randomly a  value  s  =  1, ... q,  which  is  assigned  to all  its spins.  This \nprocedure  defines  one Monte Carlo step  and  needs  to be iterated in order  to obtain \nthermodynamic averages. \nAt  temperatures  where  large  regions  of correlated spins  occur,  local  methods  (e. g. \nMetropolis), which flip  one spin at a time, become very slow.  The SVl  method over(cid:173)\ncomes this difficulty by flipping large clusters of aligned spins simult.aneously.  Hence \nthe  SW method exhibits much  smaller autocorrelation  times than local  methods . \nThe strong  connection  between  the  SW clusters  and the ordering properties  of the \nPot.ts  spins  is  manifested  in  the  relation \n\n-6-- (<1- 1)710+  1 \n\n_\".,8)  -\n\nq \n\n(4) \n\n\f420 \n\nM.  BLATI, S.  WISEMAN, E. DOMANY \n\nwhere  nij  =  1  whenever  Si  and  Sj  belong  to  the  same  SW-cluster  and  nij  =  0 \notherwise.  Thus, nij is the probability that Si  and Sj  belong to the same SW-cluster. \nThe r.h.s.  of (4)  has  a  smaller variance  than its  l.h.s., so  that  the  probabilities nij \nprovide an improved estimator of the spin spin  correlation function. \n\n4  Locating the super-paramagnetic phase \n\nIn  order  to  locate  the  temperature  range  in  which  the  system  IS  III  the  super(cid:173)\nparamagnetic phase  we  measure  the  susceptibility  of the  system  which  is  propor(cid:173)\ntional to the variance of the magnetization \nN -\n\n2 \n\nX =  T (m 2  - m  )  . \n\n(5) \n\nThe magnetization, m , is  defined  as \n\nqNmax/N -1 \nm= - - - - - -\n\nq-1 \n\n(6) \n\nwhere  NJ.'  is  the  number of spins with  the  value J.l. \nIn  the  ferromagnetic  phase  the  fluctuations  of  the  magnetization  are  negligible, \nso  the  susceptibility,  X,  is  small.  As  the  temperature  is  raised,  a  sudden  in(cid:173)\ncrease  of the  susceptibility  occurs  at  the  transition  from  the  ferromagnetic  to  the \nsuper-paramagnetic  phase.  The  susceptibility  is  non-vanishing  only  in  the  super(cid:173)\nparamagnetic phase,  which  is  the  only  phase  where  large fluctuations  in  the  mag(cid:173)\nnetization can occur.  The point where the susceptibility vanishes again is  an upper \nbound  for  the  transition  temperature  from  the  super-paramagnetic  to the  param(cid:173)\nagnetic phase. \n\n5  The clustering procedure \n\nOur method consists of two main steps.  First we  identify the range of temperatures \nwhere  the clusters  may be observed  (that corresponding to the super-paramagnetic \nphase)  and  choose  a  temperature  within  this  range.  Secondly,  the  clusters  are \nidentified  using  the  information contained  in  the  spin  spin  correlation function  at \nthis temperature.  The procedure is summarized here,  leaving discussion  concerning \nthe  choice  of the  parameters to a  later stage. \n\n(a)  Assign  to each  point  Xi  a  q-state  Potts spin  variable  Si.  q  was  chosen  equal  to \n20  in  the example that we  present  in  this work. \n\n(b)  Find all the pairs of points having mutual neighborhood value K.  We set K  =  10. \n\n(c)  Calculate the strength  of the interactions using equation  (3). \n\n(d)  Use  the SW procedure  with the Hamiltonian (1)  to calculate the susceptibility X \nfor  various temperatures.  The transition temperature from the paramagnetic phase \ncan  be  roughly estimated by  Tc  ~ 410;(1~A)' \n\n_  1 \n\n(e)  Identify  the  range  of temperatures  of non-vanishing  X  (the  super-paramagnetic \nphase).  Identify  the  temperature Tmax  where  the susceptibility  X  is  maximal, and \nthe temperature Tvanish,  where  X  vanishes at the high  temperature side.  The opti(cid:173)\nmal temperature to identify the clusters lies  between  these  two temperatures.  As  a \nrule  of thumb we  chose  the  \"clustering temperature\"  Tcltl~  =  Tvan .. ~+Tma.r  but  the \nresults  depend  only  weakly  on  Tclu~, as  long as  T cltls  is  in  the super-paramagnetic \nrange,  Tmax  < Tcltl~  < Tvani~h. \n\n\fClustering Data through  an  Analogy to the Potts  Model \n\n421 \n\n(f)  At the clustering temperature Tclu s ,  estimate the spin spin correlation,  o s \"s J '  for \nall  neigh boring pairs of points  Xi  and  Xj,  using  (4) . \n\n(g)  Clusters  are  identified  according  to  a  thresholding  procedure.  The  spin  spin \ncorrelation  function  03. ,3J  of points  Xi  and  Xj  is  compared  with  a  threshold,  ();  if \nOS,, 3J  > ()  they  are  defined  as  \"friends\".  Then  all  mutual friends  (including fl'iends \nof friends , etc)  are assigned  to the same cluster.  We  chose  ()  = 0.5 . \n\nIn order to show  how this algorithm works,  let us  consider the distribution of points \npresented  in  figure  1.  Because  of the overlap of the larger sparse  rectangle  with the \nsmaller  rectangles,  and  due  to  statistical  fluctuations,  the  three  dense  rectangles \nactually contain 883,  874  and 863 points. \nGoing  through steps  (a)  to  (d)  we  obtained  the  susceptibility  as  a  function  of the \ntemperature as presented in figure 3.  The susceptibility X is maximal at T max  =  0.03 \nand  vanishes at Tvanish  =  0.13 .  In figure  1 we  present  the clusters obtained accord(cid:173)\ning to steps (f)  and (g) at Tclus  =  0.08.  The size of the largest clusters in  descending \norder is  900 , 894, 877,  2 and all  the  rest  are  composed of only one  point.  The three \nbiggest  clusters correspond  to the clusters  we  are looking for,  while the background \nis  decomposed  into clusters  of size  one. \n\n0.035 \n\n0030 \n\n0025 \n\n0020 \n\n0015 \n\n0 0 10 \n\n0 005 \n\n0000 \n\n0 .00 \n\n0.02 \n\n0.04 \n\n0.06 \n\n0 ,08 \n\nT \n\n0.10 \n\n012 \n\no.te \n\n016 \n\nFigure  3:  The susceptibil(cid:173)\nity  density  x;;.  as  a  func(cid:173)\ntion  of t.he  t.emperature. \n\nLet  us  discuss  the effect  of the  parameters on  the  procedure.  The number of Potts \nstates,  q, determines mainly the sharpness of the transition and the critical temper(cid:173)\nature.  The higher  q,  the sharper  the  transition .  On  the other  hand, it is  necessary \nto perform more statistics (more SW sweeps)  as  the value of q increases .  From our \nsimulations, we  conclude that the influence of q is very  weak .  The maximal number \nof neighbors,  f{,  also affects the results very  little; we  obtained quite similar results \nfor  a  wide  range of f{  (5  ~ f{  ~ 20). \nNo  dramatic changes  were  observed  in  the  classification,  when  choosing  clustering \ntemperatures Tc1u3  other  than that suggested  in  (e).  However  this  choice  is  clearly \nad-hoc  and a  better choice  should  be found.  Our  method does  not  provide  a  natu(cid:173)\nral  way  to  choose  a  threshold  ()  for  the  spin  spin  correlation  function.  In  practice \nthough,  the  classification  is  not  very  sensitive  to  the  value  of (),  and  values  in  the \nrange  0.2  < ()  < 0.8  yield  similar results.  The  reason  is  that  the  frequency  distri(cid:173)\nbution  of the  values  of the  spin  spin  correlation  function  exhibit.s  t.wo  peaks,  one \nclose  to  1  and  the  other  close  to  1,  while  for  intermediate  values  it  is  verv  close \nt.o  zero.  In  figure  (4)  we  present  the  average  size  of the  largest  S\\V  cluster  as  a \nfunction  of the  temperature ,  along  with  the  size  of the  largest  cluster  obtained  by \nthe  thresholding procedUl'e  (described  in  (7))  using  three  different  threshold  values \n()  =  0.1, 0 . .5 , o .~).  Not.e  the agreement.  between  the largest  clust.er  size  defined  by  t.he \nthreshold e =  0.5 and the average size of the  largest SW cluster for  all  t.emperatures \n(This agreement holds for  the smaller clusters as well) .  It support.s our thresholding \nprocedure  as  a  sensible  one  at all  temperatUl'es. \n\nq \n\nv \n\n\f422 \n\nM.  BLATT, S.  WISEMAN, E.  DOMANY \n\nFigure  4:  Average  size  of \nthe  largest  SW  cluster  as \na  function  of the  temper(cid:173)\nature ,  is  denoted  by  the \nsolid  line.  The  triangles, \nx's and squares denote the \nsize  of  the  largest  cluster \nobtained  with  thresholds \n()  =  0.2,  0.5  and  0.9  re(cid:173)\nspectively. \n\n500 \n\no~~~~~~~~~~~~~ \n0.00  0.02  0.04  0.06  0.08  0.10  0.12  0.14  0.16 \n\nT \n\n6  Discussion \n\nOther  methods that  were  proposed  previously,  such  as  Fukunaga's  (1990) , can  be \nformulated  as  a  Metropolis  relaxation  of a  ferromagnetic  Potts  model  at  T  = O. \nThe  clusters  are  then  determined  by  the  points having  the same spin  value  at  the \nlocal minima of the energy  at which the relaxation process  terminates.  Clearly this \nprocedure  depends strongly on the initial conditions.  There is  a high probability of \ngetting stuck in a  metastable state that does  not correspond  to the  desired  answer. \nSuch a  T = 0 method does  not provide any way to distinguish between  \"good\"  and \n\"bad\"  metastable states.  We  applied  Fukunaga's method on  the  data set  of figure \n(1)  using  many  different  initial  conditions.  The  right  answer  was  never  obtained. \nIn all  runs,  domain walls that broke a  cluster  into two or more parts appeared. \nOur method generalizes  Fukunaga's method by introducing a  finite  temperature at \nwhich  the  division  into  clusters  is  stable.  In  addition,  the  SW  dynamics are  com(cid:173)\npletely insensitive to the initial conditions and extremely efficient . \nWork in  progress shows  that our method is  especially suitable for  hierarchical  clus(cid:173)\ntering.  This is done by identifying clusters at several temperatures which are chosen \naccording  to features  of the susceptibility  curve.  In  particular  our  method  is  suc(cid:173)\ncessful  in dealing with  \"real  life\"  problems such  as  the Iris  data and  Landsat  data. \n\nAcknowledgments \n\nWe  thank 1.  Kanter for  many useful  discussions.  This research  has  been  supported \nby  the  US-Israel  Bi-national  Science  Foundation  (BSF) ,  and  the  Germany-Israel \nScience  Foundation  (GIF). \n\nReferences \n\nJ .M.  Buhmann  and  H.  Kuhnel  (1993);  Vector  quantization  with  complexity  costs, \nIEEE  Trans.  Inf.  Theory  39,  1133. \nK.  Fukunaga (1990);  Introd.  to  statistical Pattern  Recognition,  Academic Press. \n\nK.  Rose , E.  Gurewitz,  and G.C . Fox (1993);  Constrained  clustering  as  an  optimiza(cid:173)\ntion  method, IEEE Trans on  Patt.  Anal.  and  Mach.  Intel.  PAMI 15, 785. \n\nS. Wang and R.H . Swendsen  (1990);  Cluster Monte  Carlo  alg.,  Physica A  167,565. \n\nF.Y.  Wu  (1982) ,  The  Potts  model,  Rev  Mod Phys,  54, 235. \nA.L.  Yuille and J.J . Kosowsky  (1994);  Statistical  algorithms  that  converge,  Neural \nComputation 6,  341  (1994). \n\n\f", "award": [], "sourceid": 1092, "authors": [{"given_name": "Marcelo", "family_name": "Blatt", "institution": null}, {"given_name": "Shai", "family_name": "Wiseman", "institution": null}, {"given_name": "Eytan", "family_name": "Domany", "institution": null}]}