{"title": "A Neural Network for Feature Extraction", "book": "Advances in Neural Information Processing Systems", "page_first": 719, "page_last": 726, "abstract": null, "full_text": "A Neural Network for Feature Extraction \n\n719 \n\nA  Neural Network for  Feature  Extraction \n\nNathan Intrator \n\nDiv.  of Applied  Mathematics,  and \n\nCenter for  Neural Science \n\nBrown University \n\nProvidence,  RI 02912 \n\nABSTRACT \n\nThe  paper suggests a  statistical framework for  the  parameter esti(cid:173)\nmation  problem associated  with  unsupervised  learning in  a  neural \nnetwork, leading to an exploratory projection pursuit network that \nperforms  feature  extraction, or  dimensionality  reduction. \n\n1 \n\nINTRODUCTION \n\nThe search  for  a  possible  presence  of some  unspecified  structure  in  a  high  dimen(cid:173)\nsional  space  can  be  difficult  due  to  the  curse  of dimensionality  problem,  namely \nthe  inherent  sparsity  of high  dimensional  spaces.  Due  to  this  problem,  uniformly \naccurate  estimations  for  all  smooth  functions  are  not  possible  in  high  dimensions \nwith  practical sample  sizes  (Cox,  1984,  Barron,  1988). \n\nRecently, exploratory projection pursuit (PP) has been considered (Jones, 1983) as a \npotential method for  overcoming the curse of dimensionality problem (Huber, 1985), \nand new  algorithms were suggested  by  Friedman (1987),  and by Hall  (1988,  1989). \nThe  idea  is  to  find  low  dimensional  projections  that  provide  the  most  revealing \nviews  of the  full-dimensional  data  emphasizing  the  discovery  of nonlinear  effects \nsuch as clustering. \n\nMany of the  methods of classical multivariate analysis  turn  out  to  be special  cases \nof PP  methods.  Examples  are  principal  component  analysis,  factor  analysis,  and \ndiscriminant  analysis.  The various PP methods differ  by the  projection index opti(cid:173)\nmized. \n\n\f720 \n\nIntrator \n\nNeural networks seem promising for feature extraction, or dimensionality reduction, \nmainly because of their powerful parallel  computation.  Feature detecting functions \nof neurons have been studied in the past two decades (von der Malsburg,  1973,  Nass \net al.,  1973, Cooper et aI.,  1979, Takeuchi and Amari,  1979).  It has also been shown \nthat a  simplified  neuron  model  can  serve  as  a  principal component  analyzer  (Oja, \n1982). \n\nThis  paper  suggests  a  statistical framework  for  the  parameter  estimation  problem \nassociated with unsupervised learning in a neural network, leading to an exploratory \nPP  network  that  performs  feature  extraction,  or  dimensionality  reduction,  of the \ntraining  data set.  The formulation,  which  is  similar  in  nature  to  PP,  is  based  on \na  minimization  of a  cost  function  over  a  set  of parameters,  yielding  an  optimal \ndecision  rule  under  some  norm.  First,  the  formulation  of a  single  and  a  multiple \nfeature extraction are  presented.  Then a  new  projection index  (cost function)  that \nfavors  directions  possessing  multimodality,  where  the  multimodality  is  measured \nin  terms  of the  separability  property  of the  data,  is  presented.  This  leads  to  the \nsynaptic  modification  equations  governing  learning  in  Bienenstock,  Cooper,  and \nMunro (BCM) neurons (1982).  A network is presented based on the multiple feature \nextraction formulation,  and both,  the linear  and nonlinear  neurons  are  analysed. \n\nSINGLE FEATURE EXTRACTION \n\n2 \nWe  associate  a  feature  with  each  projection  direction.  With  the  addition  of a \nthreshold function  we  can  say  that  an  input  posses  a  feature  associated  with  that \ndirection if its  projection  onto that direction  is larger  than  the  threshold.  In  these \nterms,  a  one dimensional projection would be a  single feature  extraction. \n\nThe  approach  proceeds  as  follows:  Given  a  compact  set  of parameters,  define  a \nfamily  of loss functions,  where the loss  function  corresponds  to a  decision  made by \nthe  neuron  whether  to fire  or  not  for  a  given  input.  Let  the  risk  be  the  averaged \nloss  over  all  inputs.  Minimize  the  risk  over  all  possible  decision  rules,  and  then \nminimize the risk over the parameter set.  In case the risk does not yield a meaningful \nminimization problem, or when the parameter set over which the minimization takes \nplace can be restricted by some a-priori knowledge, a  penalty, i.e.  a  measure on the \nparameter set,  may be added  to the  risk. \nDefine the decision problem (11, Fo, P, L, A), where 11  = (x(1), ... , x(n\u00bb),  x(i)  E R N , \nis a  fixed  set of input vectors,  (11, Fo, P) the corresponding  probability space,  A = \n{O,  I}  the  decision  space,  and  {Le }eEBM,  Le  :  11  x  A  t---+  R  is  the  family  of loss \nfunctions.  BM  is  a  compact  set  in  RM.  Let  1)  be  the  space  of all  decision  rules. \nThe risk  Re  : 1) t---+  R, is  given by: \n\nn \n\nRe(c5)  = L P(x(i\u00bb)Le(x(i), c5(x(i))). \n\ni=l \n\nFor a  fixed  8,  the  optimal decision  c5e  is  chosen  so  that: \n\n(2.1) \n\n(2.2) \n\n\fA Neural Network for Feature Extraction \n\n721 \n\nSince  the  minimization  takes  place  over  a  finite  set,  the  minimizer  exists.  In  par(cid:173)\nticular,  for  a  given  XCi)  the  decision  88(x(i\u00bb)  is  chosen  so  that  L8(X(i),88(x(i\u00bb))  < \nL8(x(i), 1- 88(x(i\u00bb)). \nNow  we find  an optimal B that minimizes  the risk,  namely, B will  be such that: \n\n(2.3) \n\nThe minimum  with respect  to f}  exits since  BM  is compact. \n\nR8(88) becomes  a  function  that  depends only  on f},  and when f}  represents  a  vector \nin  RN,  R8  can be  viewed  as  a  projection  index. \n\n3  MULTI-DIMENSIONAL FEATURE EXTRACTION \nIn this case we have a  single layer network of interconnected units, each performing \na  single feature extraction.  All units receive  the same input and the interaction be(cid:173)\ntween the units is via lateral inhibition.  The formulation is similar  to single feature \nextraction,  with  the  addition  of interaction  between  the  single  feature  extractors. \nLet  Q  be  the  number of features  to  be  extracted  from  the  data.  The  multiple de(cid:173)\ncision  rule  88  = (8~1), ... ,8~Q\u00bb)  takes  values  in  A  = {0,1}Q.  The  risk  of node  k \nis  given  by:  R~k)(8) =  l::=l P(x(i\u00bb)L~k)(x(i), 8(k)(x(i\u00bb)),  and  the  total  risk  of the \nnetwork is  R8(8)  =  l:~=l R~k)(8). Proceeding as before,  we  can  minimize  over  the \ndecision  rules  8 to get  88 ,  and then minimize  over  f}  to get B,  as in equation  (2.3). \nThe  coupling  of the  equations  via  the  inhibition,  and  the  relation  between  the \ndifferent  features  extracted is  exhibited in  the loss  function  for  each  node  and will \nbecome clear  through  the  next  example. \n\n4  FINDING  THE  OPTIMAL  f)  FOR  A  SPECIFIC  LOSS \n\nFUNCTION \n\n4.1  A  SINGLE BCM  NEURON - ONE  FEATURE EXTRACTION \n\nIn this section,  we  present  an exploratory  PP method with a  specific  loss function. \nThe  differential  equations  performing  the  optimization  turn  out  to  be  a  good  ap(cid:173)\nproximation of the low governing synaptic weight  modification in the  BCM  theory \nfor  learning  and  memory  in  neurons.  The  formal  presentation  of the  theory,  and \nsome  theoretical analysis is  given  in  (Bienenstock,  1980,  Bienenstock et  al.,  1982), \nmean  field  theory  for  a  network  based  on  these  neurons  is  presented  in  (Scofield \nand  Cooper,  1985,  Cooper  and  Scofield,  1988),  more  recent  analysis  based  on  the \nstatistical viewpoint  is  in  (Intrator  1990),  computer  simulations and the  biological \nrelevance  are discussed in (Saul et  al.,  1986,  Bear et  al.,  1987,  Cooper et al.,  1988). \n\nWe  start  with  a  short  review  of  the  notations  and  definitions  of  BCM  theory. \nConsider  a  neuron  with  input  vector  x  =  (Xl, ... , XN),  synaptic  weights  vector \nm = (ml' ... , mN),  both in RN ,  and  activity  (in  the linear  region)  c = X . m. \n\n\f722 \n\nIntrator \n\nDefine  em = E[(x\u00b7 m)2], \u00a2(e, em) = e2 - jeem , 4>(e,  em) = e2 - ~eem. The input \nx, which is a  stochastic process, is assumed to be of Type II t.p  mixing,  bounded, and \npiecewise  constant.  The  t.p  mixing  property  specifies  the  dependency  of the  future \nof the  process  on its  past.  These  assumptions  are  needed  for  the approximation of \nthe  resulting deterministic equation  by a  stochastic one  and  are  discussed in  detail \nin  (Intrator,  1990).  Note  that  e  represents  the  linear  projection  of x  onto m,  and \nwe  seek  an  optimal projection in some sense. \nThe  BCM  synaptic  modification  equations  are  given  by:  m =  JL(t)4>(x  . m, em)x, \nm(O) = mo,  where JL(t)  is a  global modulator which is assumed to take into account \nall  the  global  factors  affecting  the  cell,  e.g.,  the  beginning  or  end  of the  critical \nperiod,  state of arousal,  etc. \nRewriting  the  modification  equation  as  m =  JL(t)(x  . m)(x  . m  - ~!1m)X,  we  see \nthat  unlike  a  classical  Hebb-Stent  rule,  the  threshold  !1m  is  dynamic.  This  gives \nthe  modification  equation  the  desired  stability,  with  no  extra  conditions  such  as \nsaturation of the  activity,  or  normalization  of II  m  II,  and  also  yields  a  statistically \nmeaningful optimization. \nReturning  to  the  statistical  formulation,  we  let  !1  = m  be  the  parameter  to  be \nestimated according to the above formulation and define an appropriate loss function \ndepending on the cell's decision  whether to fire  or not.  The loss function  represents \nthe  intuitive  idea that  the  neuron  will  fire  when  its  activity  is  greater  than  some \nthreshold,  and  will  not  otherwise.  We  denote  the  firing  of the  neuron  by  a  =  1. \nDefine  K  = -JL JJe ... \u00a2(s, em)ds. Consider  the following loss function: \n\n... \n\nL8(X, a)  =  Lm(x, a)  = \n\n.l(z.m)  A( \n\n... .l{z.m)  A \n\n4>  s, em  s, \n\n)d \n\n4>(s, em)ds, \n\n-JL  e \nK  - JL  e... \n-JL  e... \nK \n\n- JL  0... \n\n.l(z.m)  A ( \n\n4>  s,  em  s, \n\n)d \n\n.l{z.m)  A ( \n\n4>  s, em  s, \n\n)d \n\n_  m, \n\n(x\u00b7 m)  > e \na=1 \n(x\u00b7 m)  < em,  a=1 \n(x\u00b7 m)  ~ em,  a=O \n(x\u00b7 m)  > em,  a=O \n\nIt follows  from  the  definition of L8  and from  the  definition  of 68  in  (2.2)  that \n\nLm(x, 6m) = -JL  {(z.m) \u00a2(s, em)ds = - JL {(x. m)3 - E[(x . m)2](x . m)2} \n\nJe ... \n\n3 \n\n(4.1) \n\n( 4.2) \n\nThe  above  definition  of the  loss  function  suggests  that  the  decision  of a  neuron \nwhether  to fire  or not is  based on  a  dynamic threshold  (x  . m) > em.  It turns out \nthat  the  synaptic  modification  equations  remain  the  same  if the  decision  is  based \non  a  fixed  threshold.  This  is  demonstrated  by  the  following  loss  function,  which \nleads to the same risk  as in equation  (4.3):  K  =  -JL Joje ...  \u00a2(s, em)ds, \n\nL8(X, a) = Lm(x, a) = \n\ns, \n\n)d \n\n4>  s, em \n\n((z.m)  A( \nJL  J~z .m) \u00a2(s, em)ds, \n((z.m)  A( \n\n4>  s, em  s, \n(z.m)  A  _ )  \n\n-JL Jo \nK  -\n-JL Jo \nK  - JL  Jo \n\n4>(s,  em  ds, \n\n)d \n\n(x  . m)  ~ 0,  a =  1 \n(x\u00b7 m)  < 0,  a =  1 \n(x\u00b7 m)  ~ 0,  a =  0 \n(x. m)  > 0,  a =  0 \n\n( 4.1') \n\n\fA Neural Network for Feature Extraction \n\n723 \n\nThe risk  is given  by: \n\nThe  following  graph  represents  the  \u00a2  function  and  the  associated  loss  function \nLm(x, 6m) of the  activity c. \n\nTHE 4>  FUNCTION \n\nTHE LOSS  FUNCTION \n\n(4.3) \n\nFig.  1:  The Function \u00a2  and the  Loss  Functions for  a  Fixed m  and em. \n\nFrom the graph of the loss function it follows  that for  any fixed  m  and em,  the loss \nis small for  a  given input  x,  when  either  x  . m  is close  to zero or negative,  or when \nx  . m  is larger than em.  This suggests,  that  the  preferred directions for  a  fixed  8m \nwill  be  such  that the  projected single  dimensional distribution  differs from  normal \nin the center of the distribution, in the sense that it has a  multi-modal distribution \nwith a  distance  between the  two peaks larger than 8m \u2022  Rewriting (4.3)  we  get \n\nRe(6e)  __ !!..  E[(x\u00b7 m)3]  _  1 \n\nE2[(x . m)2]  -\n\n3 {E2[(x . m)2] \n\n}. \n\n(4.4) \n\nThe term E[(x.m)3]/E2[(x.m)2] can be viewed as some measure of the skewness of \nthe  distribution,  which  is  a  measure  of deviation from  normality  and  therefore  an \ninteresting direction  (Diaconis  and  Friedman,  1984),  in  accordance  with  Friedman \n(1987)  and  Hall's  (1988,  1989)  argument  that  it  is  best  to  seek  projections  that \ndiffer  from  the  normal in the  center of the  distribution rather than in the  tails. \n\nSince  the  risk  is  continuously  differentiable,  its  minimization  can  be  done  via  the \ngradient  descent  method with respect  to  m,  namely: \n\n( 4.5) \n\nNotice  that  the  resulting  equation  represents  an  averaged  deterministic  equation \nof the  stochastic  BCM  modification  equations.  It  turns  out  that  under  suitable \nconditions on  the mixing of the input  x  and the global function IL,  equation  (4.5)  is \na  good approximation of its stochastic version. \n\nWhen  the  nonlinearity  of the  neuron  is  emphasized,  the  neuron's  activity  is  then \ndefined  as  c  =  0'( X  \u2022  m),  where  0'  usually  represents  a  smooth  sigmoidal  function. \nem  is  then  defined  as  E[0'2(x  . m)],  and  the  loss  function  is  similar  to  the  one \ngiven by equation (4.1)  except that (x\u00b7 m)  is replaced by O'(x, m).  The gradient of \n\n\f724 \n\nIntrator \n\nthe risk is  given  by:  -VTn.Rnt(8m) =  J-LE[\u00a2(O'(x, m), E>m ) 0\" x],  where  0\"  represents \nthe  derivative  of 0'  at  the  point  (x . m).  Note  that  0'  may represent  any  nonlinear \nfunction,  e.g.  radial  symmetric kernels. \n\n4.2  THE NETWORK - MULTIPLE  FEATURE EXTRACTION \n\nIn  this  case  we  have  Q  identical  nodes,  which  receive  the  same  input  and  inhibit \neach  other.  Let  the  neuronal  activity  be  denoted  by  Ck  =  X  \u2022  mk.  We  define  the \ninhibited activity  Ck  =  Ck  -\n11 Eitk ci'  and  the  threshold  e~ =  E[cil.  In  a  more \ngeneral case,  the inhibition may be defined  to take into account the spatial location \nof adjacent  neurons,  namely,  Ck  =  Ei Aikci,  where  Aik  represents  different  types \nof inhibitions, e.g.  Mexican hat.  Since  the following  calculations  are valid for  both \nkinds of inhibition we  shall introduce only  the  simpler one. \n\nThe loss function is similar to the one defined in a  single feature extraction with the \nexception that the activity C = X\u00b7 m  is replaced by C.  Therefore the risk for  node k is \ngiven by:  Rk  = -~{E[c~] - (E[ci])2}, and the total risk is given by R = E~=l Rk. \nThe gradient  of R  is  given  by: \n\n8R  =  -J-L[I-11(Q - l)lE[\u00a2(ck,e~)x]. \n8mk \n\n(4.6) \n\nEquation  (4.6)  demonstrates the ability of the  network to  perform exploratory pro(cid:173)\njection pursuit in  parallel,  since  the  minimization  of the risk  involves minimization \nof nodes  1, ... , Q,  which  are loosely  coupled. \n\nThe  parameter  11  represents  the  amount  of lateral  inhibition  in  the  network,  and \nis  related  to  the  amount  of correlation  between  the  different  features  sought  by \nthe  network.  Experience  shows  that  when  11  ~ 0,  the  different  units  may  all  be(cid:173)\ncome  selective  to the  simplest feature  that  can be  extracted from  the  data.  When \nl1(Q - 1)  ~ 1,  the  network becomes selective  to those inputs that are very far  apart \n(under  the  l2  norm),  yielding  a  classification  of a  small  portion  of the  data,  and \nmostly  unresponsiveness  to  the  rest  of the  data.  When 0 < 11( Q - 1) < 1,  the  net(cid:173)\nwork becomes responsive  to substructures that may be common to several different \ninputs,  namely  extract invariant features  in  the  data.  The  optimal  value  of 11  has \nbeen estimated by data driven  techniques. \n\nWhen  the  non  linearity  of the  neuron  is  emphasized  the  activity  is  defined  (as  in \nthe single  neuron case)  as Ck  =  0'( X  \u2022 mk)'  Ck,  e~, and Rk  are defined as  before.  In \nthis case :!: = -l1O\"(X' mi)x,  :!kk  = O\"(x\u00b7 mk)x,  and equation  (4.6)  becomes: \n\n4.3  OPTIMAL  NETWORK  SIZE \n\nA  major  problem  in  network solutions  to  real  world  problems  is  optimal  network \nsize.  In our  case,  it is  desirable  to  try  and extract as  many features  as possible  on \n\n\fA Neural Network for Feature Extraction \n\n725 \n\none  hand,  but  it is clear  that  too  many neurons in  the network will  simply inhibit \neach  other,  yielding  sub-optimal  results.  The following  solution  was  adopted:  We \nreplace  each  neuron  in  the  network  with  a  group  of neurons  which  all  receive  the \nsame  input,  and  the  same  inhibition  from  adjacent  groups.  These  neurons  differ \nfrom  one  another only in  their initial synaptic weights.  The output of each neuron \nis  replaced  by  the  average  group  activity.  Experiments  show  that  the  resulting \nnetwork is more robust to noise and outliers in the data.  Furthermore, it is observed \nthat  groups  that  become  selective  to  a  true  feature  in  the  data,  posses  a  much \nsmaller  inter-group  variance  of their  synaptic  weight  vector  than  those  which  do \nnot  become  responsive  to  a  coherent  feature.  We  found  that  eliminating  neurons \nwith  large  inter-group  variance  and  retraining  the  network,  may  yield  improved \nfeature  extraction properties. \n\nThe  network has  been  applied  to  speech  segments, in  an  attempt  to extract  some \nfeatures  from  CV  pairs of isolated  phonemes  (Seebach  and Intrator,  1988). \n\n5  DISCUSSION \nThe PP method based on the BCM modification function, has been found capable of \neffectively  discovering non linear data structures in  high dimensional spaces.  Using \na  parallel  processor  and  the  presented  network topology,  the  pursuit  can  be  done \nfaster  than in  the traditional serial  methods. \n\nThe  projection  index is  based  on  polynomial  moments,  and  is  therefore  computa(cid:173)\ntionally  attractive.  When  only  the  nonlinear  structure in  the  data is  of interest,  a \nsphering transformation  (Huber,  1981,  Friedman, 1987),  can be applied first  to the \ndata for  removal of all the location, scale, and correlational structure from the data. \n\nWhen compared with other PP methods,  the highlights of the presented method are \ni) the projection index concentrates on directions where the separability property as \nwell as the non-normality of the data is large, thus giving rise to better classification \nproperties; ii) the degree of correlation between the directions, or features extracted \nby the network can  be regulated  via the  global inhibition,  allowing some  tuning of \nthe network to different  types of data for  optimal results;  iii) the pursuit is done on \nall the directions  at  once  thus leading  to  the  capability  of finding  more interesting \nstructures  than  methods that  find  only  one  projection  direction  at a  time.  iv)  the \nnetwork's structure suggests a  simple method for  size-optimization. \n\nAcknowledgements \n\nI  would like  to  thank  Professor  Basilis  Gidas for  many fruitful  discussions. \n\nSupported  by  the  National  Science  Foundation,  the  Office  of Naval  Research,  and \nthe Army  Research  Office. \n\nReferences \nBarron  A.  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