{"title": "Directional-Unit Boltzmann Machines", "book": "Advances in Neural Information Processing Systems", "page_first": 172, "page_last": 179, "abstract": null, "full_text": "Directional-Unit  Boltzmann Machines \n\nRichard S.  Zemel \n\nComputer Science  Dept. \n\nUniversity of Toronto \n\nToronto,  ONT M5S  lA4 \n\nChristopher K.  I.  Williams \n\nComputer Science  Dept. \nUniversity  of Toronto \n\nToronto, ONT  M5S  lA4 \n\nMichael C.  Mozer \n\nComputer Science  Dept. \nUniversity of Colorado \nBoulder,  CO  80309-0430 \n\nAbstract \n\nWe  present  a  general  formulation  for  a  network  of stochastic  di(cid:173)\nrectional units.  This formulation is  an extension of the Boltzmann \nmachine in which  the units are  not  binary, but take on values in a \ncyclic  range,  between  0  and 271'  radians.  The state of each  unit  in \na  Directional-Unit  Boltzmann  Machine  (DUBM)  is  described  by  a \ncomplex variable, where  the phase component specifies  a direction; \nthe  weights  are  also  complex variables.  We  associate  a  quadratic \nenergy  function,  and  corresponding  probability,  with  each  DUBM \nconfiguration.  The  conditional  distribution  of a  unit's  stochastic \nstate is  a  circular  version of the Gaussian probability distribution, \nknown  as  the  von  Mises  distribution.  In  a  mean-field  approxima(cid:173)\ntion  to  a  stochastic  DUBM,  the  phase  component of a  unit's state \nrepresents  its mean direction,  and the magnitude component spec(cid:173)\nifies  the  degree  of certainty  associated  with  this  direction.  This \ncombination of a  value  and  a  certainty  provides  additional  repre(cid:173)\nsentational power  in a  unit.  We  describe  a  learning algorithm and \nsimulations that demonstrate  a  mean-field  DUBM'S  ability to learn \ninteresting  mappings. \n\nMany  kinds  of information  can  naturally  be  represented  in  terms  of angular,  or \ndirectional, variables.  A circular range forms a suitable representation for  explicitly \ndirectional  information,  such  as  wind  direction,  as  well  as  for  information  where \nthe  underlying  range  is  periodic,  such  as  days  of the  week  or  months of the  year. \nIn  computer  vision,  tangent  fields  and optic flow  fields  are  represented  as fields  of \noriented line segments, each of which can be described by a magnitude and direction. \nDirections  can  also  be  used  to represent  a set  of symbolic labels,  e.g.,  object  label \nA  at  0,  and  object  label  B  at 71'/2  radians.  We  discuss  below  some  advantages  of \nrepresenting  symbolic labels with  directional units. \n\n172 \n\n\fDirectional-Unit Boltzmann Machines \n\n173 \n\nThese  and  many  other  phenomena  can  be  usefully  encoded  using  a  directional \nrepresentation-a  polar  coordinate  representation  of complex values  in  which  the \nphase parameter indicates a direction between 0 and 27r  radians.  We have devised a \ngeneral formulation of networks of stochastic directional units.  This paper describes \na  directional-unit  Boltzmann  machine  (DUBM),  which  is  a  novel  generalization of a \nBoltzmann machine  (Ackley,  Hinton  and  Sejnowski,  1985)  in  which  the  units  are \nnot  binary,  but instead  take on directional  values between  0  and 27r. \n\n1  STOCHASTIC  DUBM \n\nA  stochastic  directional unit  takes on values  on  the  unit  circle.  We  associate  with \nunit  j  a  random  variable  Zj;  a  particular  state  of  j  is  described  by  a  complex \nnumber with magnitude one  and direction,  or phase  Tj:  Zj  =  eiTj  \u2022 \nThe  weights  of a  DUBM  also  take  on  complex  values.  The  weight  from  unit  k  to \nunit  j  is:  Wj k  = hj ke ifJ'k .  We  constrain  the  weight  matrix  W  to  be  Hermitian: \nW T  =  W*,  where  the  diagonal  elements of the  matrix are  zero,  and  the  asterisk \nindicates  the  complex  conjugate  operation.  Note  that  if the  components are  real, \nthen  W T  =  W,  which  is  a  real  symmetric matrix.  Thus,  the  Hermitian form is  a \nnatural generalization of weight  symmetry to the  complex domain. \n\nThis definition  of W  leads  to a  Hermitian quadratic form  that generalizes  the real \nquadratic form  of the  Hopfield  energy  function: \n\nE(z)  =  -1/2 z*TWz  =  -1/2  LZjZZWjk \n\nj,k \n\n(1) \n\nwhere z is the vector of the units' complex states in a particular global configuration. \nNoest  (1988)  independently  proposed  this  energy  function.  It is  similar  to  that \nused  in  Fradkin,  Huberman,  and Shenker's  (1978)  generalization of the  XY  model \nof statistical mechanics to allow  arbitary weight  phases  OJ k,  and coupled  oscillator \nmodels, e.g.,  Baldi and  Meir  (1990). \nWe  can  define  a  probability  distribution  over  the  possible  states  of  a  stochastic \nnetwork  using  the  Boltzmann factor.  In  a  DUBM,  we  can  describe  the  energy  as  a \nfunction  of the state of a  particular unit j: \n\nE(Zj = Zj)  =  -1/2 [L ZjZZWjk + L  ZkZIWkj] \n\nk \n\nk \n\nWe  define \n\nXj  =  LZkwlk \n\nk \n\nto be  the  net input  to unit  j, where  aj  and  O:j  denote  the magnitude and phase of \nx j,  respectively. \nApplying  the  Boltzmann  factor,  we  find  that  the  probability  that  unit  j  IS  10  a \nparticular state is  proportional to: \n\np(Zj = Zj)  ex  e-/3E(Zj=zj)  = e/3aj  COS(Tj-crj) \n\n(2) \n\nwhere  f3  is  the  reciprocal of the system  temperature. \n\n\f174 \n\nZemel,  Williams,  and Mozer \n\n0\u00b0 \n\n---> \n\nFigure  1:  A  circular  normal  density  function  laid  over  a  unit  circle.  The  dots \nalong  the  circle  represent  samples of the  circular  normal random variable  Zj.  The \nexpected  direction of Zj,  Tj,  is  7r /4;  rj  is its resultant  length. \n\nThis probability distribution for  a  unit's state corresponds  to a  distribution known \nas  the  von  Mises,  or  circular normal,  distribution (Mardia,  1972).  Two parameters \ncompletely  characterize  this  distribution:  a  mean  direction r  = (0,27r]  and  a  con(cid:173)\ncentration  parameter  m  > 0  that  behaves  like  the  reciprocal  of the  variance  of a \nGaussian distribution on a linear random variable.  The probability density function \nof a  circular  normal random variable  Z  is l : \n1 \n() \n27r1o  m \n\np  T;  r, m  = \n( \n\nem cos( T-T) \n\n(3) \n\n) \n\nFrom Equations 2 and 3, we see  that if a unit adopts states according to its contribu(cid:173)\ntion to the system energy,  it will  be  a  circular normal variable with  mean direction \nCl:j  and  concentration  parameter  mj  =  f3aj.  These  parameters are  directly  deter(cid:173)\nmined by  the  net input  to the unit. \n\nFigure  1 shows  a  circular  normal density  function  for  Zj,  the state of unit j.  This \nfigure  also shows  the expected  value of its stochastic  state,  which  we  define  as: \n\nYj  = < Zj  > =  rjei'yi \n\n(4) \nwhere  Ij,  the phase of Yj,  is  the mean direction and rj, the magnitude of Yj,  is the \nresultant length.  For a circular normal random variable, Ij = Tj,  and rj  =  ~~~:!j.2 \nWhen samples of Zj  are concentrated on a small arc about the mean (see  Figure 1), \nrj  will  approach  length  one.  This  corresponds  to  a  large  concentration  parameter \n(mj  = f3aj).  Conversely,  for  small  mj,  the  distribution  approaches  the  uniform \ndistribution on  the  circle,  and the resultant  length falls toward zero.  For a  uniform \ndistribution,  rj  = O.  Note  that  the  concentration  parameter  for  a  unit's  circular \norder zero.  An  integral  representation  of this  function  is  Io(m)  =  ~ J: e\u00b1mcos()d6.  It can \nIThe  normalization  factor  Io(m)  is  the  modified  Bessel  function  of the  first  kind  and \n\nbe  computed  by  numerical  routines. \nk is  h(m) =  ~ Jo7r  emcos() cos(k6)d6.  Note  that  II(m) = dlo(m)/dm. \n\n2 An  integral  representation  of the  modified  Bessel  function  of the first  kind  and order \n\n\fDirectional-Unit Boltzmann Machines \n\n175 \n\nnormal density function  is  proportional to /3,  the reciprocal of the system tempera(cid:173)\nture.  Higher temperatures will thus have the effect of making this distribution more \nuniform, just as  they  do in  a  binary-unit  Boltzmann machine. \n\n2  EMERGENT PROPERTIES  OF  A  DUBM \n\nA  network  of directional  units  as  defined  above  contains  two  important emergent \nproperties.  The  first  property  is  that  the  magnitude  of the  net  input  to  unit  j \ndescribes  the extent  to which  its various inputs  \"agree\".  Intuitively, one  can  think \nof each  component  Zk wj k  of the  sum  that  comprises  x j  as  predicting  a  phase  for \nunit j.  When the phases of these  components are equal,  the magnitude of Xj,  aj, is \nmaximized.  If these phase predictions are far apart, then they will act to cancel each \nother out,  and  produce  a  small aj.  Given  Xj,  we  can  compute  the  expected  value \nof the  output of unit  j.  The expected  direction of the  unit  roughly  represents  the \nweighted average of the phase predictions, while the resultant length is a monotonic \nfunction  of aj  and hence  describes  the  agreement between  the  various predictions. \n\nThe key  idea here  is  that  the  resultant  length  directly  describes  the  degree  of cer(cid:173)\ntainty in  the  expected  direction  of unit j.  Thus,  a  DUBM  naturally incorporates  a \nrepresentation of the system's confidence in  a value.  This ability to combine several \nsources  of evidence,  and  not  only  represent  a  value  but  also  describe  the  certainty \nof that value is  an important property  that may be  useful  in  a  variety of domains. \n\nThe  second  emergent  property  is  that  the  DUBM  energy  is  globally  rotation(cid:173)\ninvariant-E  is  unaffected  when  the  same  rotation  is  applied  to  all  units'  states \nin  the  network.  For  each  DUBM  configuration,  there  is  an equivalence  class  of con(cid:173)\nfigurations which have the same energy.  In a similar way, we find that the magnitude \nof Xj  is  rotation-invariant.  That is,  when  we  translate  the  phases  of all  units  but \none  by some phase,  the magnitude of that unit is  unaffected.  This property under(cid:173)\nlies one of the key  advantages of the representation:  both the magnitude of a unit's \nstate  as  well  as  system  energy  depend  on  the  relative  rather  than  absolute  phases \nof the  units. \n\n3  DETERMINISTIC  DUBM \n\nJust  as in  deterministic binary-unit Boltzmann machines  (Peterson  and Anderson, \n1987;  Hinton,  1989),  we  can  greatly  reduce  the  computational  time  required  to \nrun  a  large  stochastic  system  if  we  invoke  the  mean-field  approximation,  which \nstates that once the system has reached equilibrium, the stochastic variables can  be \napproximated by their mean values.  In  this approximation, the variables are treated \nas  independent,  and  the  system  probability  distribution  is  simply  the  product  of \nthe probability distributions for  the  individual units. \n\nGislen,  Peterson,  and Soderberg  (1992)  originally proposed  a  mean-field theory for \nnetworks  of  directional  (or  \"rotor\")  units,  but  only  considered  the  case  of real(cid:173)\nvalued  weights.  They  derived  the  mean-field  consistency  equations  by  using  the \nsaddle-point method.  Our approach provides an alternative, perhaps more intuitive \nderivation,  due  to the  use  of the  circular normal distribution. \n\n\f176 \n\nZemel,  Williams,  and Mozer \n\nWe  can  directly  describe  these  mean  values  based on  the circular  normal interpre(cid:173)\ntation.  We  still denote  the net  input to  a  unit j  as  Xj: \nx j  = ~ Yk W j k  = aj e  1 \n\n~  * \n\niao \n\n(5) \n\nOnce  equilibrium has  been  reached,  the state of unit j  is  Yj,  the expected  value  of \nZj  given the mean-field approximation: \n\nk \n\n(6) \n\nIn  the stochastic  as well  as  the deterministic system,  units evolve  to minimize the \nfree  energy,  F  = < E  > - T H.  The calculation of H,  the  entropy of the  system, \nfollows directly from the circular normal distribution and the mean-field approxima(cid:173)\ntion.  We  can  derive  mean-field consistency  equations for  Xj  and  Yj  by  minimizing \nthe  mean-field free  energy,  FM F,  with  respect  to each  variable independently.  The \nresulting  equations  match  the  mean-field  equations  (Equations  5  and  6)  derived \ndirectly from the circular normal probability density function.  They also match the \nspecial  case  derived  by  Gislen  et  al.  for  real-valued  weights. \n\nWe  have implemented a  DUBM  using  the mean-field  approximation.  We  solve for  a \nconsistent set of x  and y  values by performing synchronous updates of the discrete(cid:173)\ntime  approximation of the  set  of differential  equations  based  on  the  net  input  to \neach  unit j.  We  update the  x j  variables using  the following differential equation: \n\ndXj \n\n~  * \n--;It  = -Xj + ~ YkWjk \n\nk \n\n(7) \n\nwhich  has  Equation 5 as  its steady-state solution.  In  the simulations, we  use  simu(cid:173)\nlated annealing to help  find  good  minima of FM F. \n\nJust as for  the  Hopfield  binary-state network,  it can  be shown  that the free  energy \nalways  decreases  during  the  dynamical evolution  described  in  Equation  7  (Zemel, \nWilliams and  Mozer,  1992).  The equilibrium solutions  are free  energy  minima. \n\n4  DUBM  LEARNING \n\nThe units in a  DUBM  can be arranged in a variety of architectures.  The appropriate \nmethod  for  determining  weight  values  for  the  network  depends  on  the  particular \nclass of network  architecture.  In  an autoassociative network containing a single set \nof interconnected  units,  the  weights  can  be set  directly from the  training patterns. \nIf hidden  units  are  required  to perform  a  task,  then  an  algorithm for  learning  the \nweights  is  required.  We  use  an  algorithm that generalizes  the  Boltzmann machine \ntraining  algorithm  (Ackley,  Hinton  and  Sejnowski,  1985;  Peterson  and  Anderson, \n1987)  to  these  networks. \n\nAs  in the standard Boltzmann machine learning algorithm, the partial derivative of \nthe objective function  with  respect  to  a  weight  depends  on  the  difference  between \nthe partials of two mean-field free  energies:  one  when  both input and output units \nare  clamped,  and  the  other  when  only  the  input  units  are  clamped.  On  a  given \n\n\fDirectional-Unit Boltzmann Machines \n\n177 \n\ntraining case,  for  each  of these  stages  we  let  the  network  settle  to  equilibrium and \nthen  calculate  the following derivatives: \n\nOFMF/objk \nOFM F / O(}j k \n\n'Yk  + (}jk) \n-rjTk COS(-yj  -\nrjrkbjk sin('Yj  - 'Yk  + (}jk) \n\nThe learning algorithm uses these gradients to find weight values that will minimize \nthe objective over  a  training set. \n\n5  EXPERIMENTAL RESULTS \n\nWe  present  below  some illustrative  examples  to show  that an  adaptive  network  of \ndirectional units can be used in a range of paradigms, including associative memory, \ninput/output mappings, and  pattern  completion. \n\n5.1  SIMPLE AUTOASSOCIATIVE  DUBM \n\nThe first set of experiments considers a simple autoassociative DUBM,  which contains \nno  hidden  units,  and  the  units  are  fully  connected.  As  in  a  standard  Hopfield \nnetwork,  the  weights  are  set  directly  from  the  training  patterns;  they  equal  the \nsuperposition  of the outer  product  of the patterns. \n\nWe have run several experiments with simple autoassociative DUBMs.  The empirical \nresults parallel those for binary-unit autoassociative networks.  We find, for example, \nthat a network containing 30 fully interconnected units is capable of reliably settling \nfrom  a  corrupted  version  of one  of 4  stored  patterns  to  a  state  near  the  pattern. \nThese  patterns  thus  form  stable  attractors,  as  the  network  can  perform  pattern \ncompletion and clean-up from noisy inputs.  The rotation-invariance property of the \nenergy  function  allows  any  rotated  version  of a  training  pattern  to  also  act  as  an \nattractor.  The network's performance rapidly degrades for  more than 4 orthogonal \npatterns;  the  patterns  themselves  no longer  act  as  fixed-points,  and  many random \ninitial states end in states far from any stored pattern.  In addition, more orthogonal \npatterns  can  be stored  than  random  patterns.  See  Noest  (1988)  for  an  analysis  of \nthe capacity of an autoassociative  DUBM  with sparse  and asymmetric connections. \n\n5.2  LEARNING INPUT/OUTPUT MAPPINGS \n\nWe  have also used  the  mean-field DUBM  learning algorithm to learn  the  weights  in \nnetworks  containing hidden units.  We  have  experimented with  a  task  that is  well(cid:173)\nsuited to a directional representation.  There is a single-jointed robot arm, anchored \nat  a  point,  as  shown  in  Figure  2.  The  input  consists  of  two  angles: \nthe  angle \nbetween  the  first  arm segment  and  the  positive  x-axis  (A),  and  the  angle  between \nthe  two  arm  segments  (p).  The  two segments each  have  a  fixed  length,  A  and  B; \nthese  are  not explicitly given  to  the  network.  The output is  the angle  between  the \nline  connecting  the  two  ends  of the  arm  and  the  x-axis  (J.t).  This  target  angle  is \nrelated  in  a  complex,  non-linear  way  to the  input angles-the network  must learn \nto approximate the following trigonometric relationship: \n\nJ.1.  =  arctan \n\n( A sin A - B sin( A + p)  ) \nA cos A - B cos( A + p) \n\n\f178 \n\nZemel, Williams,  and Mozer \n\n/ \n\nI \n\nI \n\n,\\  ~ p, \n\n----------- -j~----------~ \n\nFigure 2:  A sample training case  for  the  robot  arm problem.  The  arm consists  of \ntwo fixed-length segments, A  and B, and is  anchored on the x-axis.  The two angles, \n,\\  and p,  are  given as  input for  each  case,  and the target  output is  the angle p,. \n\nWith  500  training cases,  a  DUBM  with  2 input units and 8 hidden  units is  able  to \nlearn  the  task so  that it can accurately  estimate p,  for  novel  patterns.  The learning \nrequires  200  iterations of a  conjugate gradient  training algorithm.  On  each  of 100 \ntesting patterns,  the  resultant  length of the output unit exceeds  .85,  and the mean \nerror  on  the  angle  is  less  than  .05  radians.  The  network  can  also  learn  the  task \nwith  as few  as  5 hidden  units,  with  a  concomitant decrease  in learning speed.  The \ncompact  nature  of this  network  shows  that  the  directional  units  form  a  natural, \nefficient  representation  for  this problem. \n\n5.3  COMPLEX PATTERN  COMPLETION \n\nOur  earlier  work  described  a  large-scale  DUBM  that  attacks  a  difficult  problem in \ncomputer  vision:  image segmentation.  In  MAGIC  (Mozer  et  al.,  1992),  directional \nvalues are used to represent alternative labels that can be assigned to image features. \nThe goal of MAGIC  is  to  learn  to assign  appropriate object  labels to  a set of image \nfeatures  (e.g.,  edge  segments)  based  on  a  set  of examples.  The  idea  is  that  the \nfeatures of a given object should have  consistent phases,  with each  object taking on \nits own phase.  The units in  the  network  are  arranged into two layers-feature and \nhidden-and the  computation proceeds  by  randomly initializing the  phases  of the \nunits in the feature layer,  and settling on a labeling through a relaxation procedure. \nThe  units  in  the  hidden  layer  learn  to  detect  spatially  local  configurations  of the \nimage features  that are labeled in a consistent manner across the training examples. \n\nMAGIC successfully learns to segment novel scenes  consisting of overlapping geomet(cid:173)\nric objects.  The emergent DUBM properties described above are essential to MAGIC'S \nability  to  perform  this  task.  The  complex  weights  are  necessary  in  MAGIC,  as  the \nweights encode  statistical  regularities  in  the  relationships  between  image features, \ne.g.,  that  two features  typically belong  to the  same object  (i.e.,  have similar phase \nvalues)  or  to  different  objects  (i.e.,  are  out  of phase).  The  fact  that  a  unit's  re(cid:173)\nsultant  length  reflects  the  certainty  in  a  phase  label  allows  the  system  to  decide \nwhich  phase  labels  to  use  when  updating  labels  of neighboring  features:  the  ini(cid:173)\ntially  random  phases  are  ignored,  while  confident  labels  are  propagated.  Finally, \nthe  rotation-invariance property  allows  the system to assign  labels to features  in  a \nmanner consistent  with  the  relationships  described  in  the  weights,  where  it  is  the \nrelative  rather  than  absolute  phases of the units  that are  important. \n\n\fDirectional-Unit Boltzmann Machines \n\n179 \n\n6  CURRENT DIRECTIONS \n\nWe  are  currently  extending  this work  in  a  number of directions.  We  are extending \nthe  definition  of a  DUBM  to  combine  binary  and  directional  units  (Radford  Neal, \npersonal  communication).  This expanded  representation  may be useful  in  domains \nwith directional data that is not present everywhere.  For example, it can be directly \napplied  to  the  object  labeling  problem  explored  in  MAGIC.  The  binary  aspect  of \nthe unit can  describe  whether  a  particular image feature  is present  or absent.  This \nmay enable the system to handle various complications, particularly labeling across \ngaps  along  the  contour  of an  object.  Finally,  we  are  applying  a  DUBM  network  to \nthe interesting and challenging problem of time-series prediction of wind directions. \n\nAcknowledgements \n\nThe  authors  thank  Geoffrey  Hinton  for  his  generous  support  and  guidance.  We \nthank Radford Neal,  Peter Dayan, Conrad Galland, Sue  Becker,  Steve  Nowlan,  and \nother  members of the  Connectionist  Research  Group  at  the  University  of Toronto \nfor  helpful comments regarding this work.  This research  was supported  by a  grant \nfrom  the  Information Technology  Research  Centre  of Ontario  to Geoffrey  Hinton, \nand NSF  Presidential Young Investigator award IRI-9058450 and grant 90-21 from \nthe James S.  McDonnell  Foundation to MM. \n\nReferences \n\nAckley,  D.  H.,  Hinton, G.  E.,  and Sejnowski, T. J.  (1985).  A learning algorithm for \n\nBoltzmann machines.  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J.  (1988).  Phasor  neural  networks.  In  Neural  Information  Processing \n\nSystems,  pages  584-591,  New  York.  AlP. \n\nPeterson,  C.  and  Anderson,  J.  R.  (1987).  A  mean field  theory  learning  algorithm \n\nfor  neural  networks.  Complex  Systems,  1:995-1019. \n\nZemel,  R.  S.,  Williams,  C.  K.  I.,  and  Mozer,  M.  C.  (1992).  Adaptive  networks  of \n\ndirectional units.  Technical  Report  CRG-TR-92-2,  University  of Toronto. \n\n\f", "award": [], "sourceid": 674, "authors": [{"given_name": "Richard", "family_name": "Zemel", "institution": null}, {"given_name": "Christopher", "family_name": "Williams", "institution": null}, {"given_name": "Michael", "family_name": "Mozer", "institution": null}]}