{"title": "Classification in Non-Metric Spaces", "book": "Advances in Neural Information Processing Systems", "page_first": 838, "page_last": 846, "abstract": null, "full_text": "Classification in Non-Metric Spaces \n\nDaphna Weinshall l ,2  David W.  Jacobs l  Yoram  Gdalyahu2 \n\n1 NEC  Research  Institute, 4  Independence  Way, Princeton,  NJ  08540,  USA \n2Inst.  of Computer Science,  Hebrew  University  of Jerusalem,  Jerusalem  91904 , Israel \n\nAbstract \n\nA  key question in vision  is  how to represent our knowledge  of previously \nencountered objects to classify new ones.  The answer depends on how we \ndetermine  the similarity  of two  objects.  Similarity  tells  us how  relevant \neach previously seen object is in determining the category to which a new \nobject belongs.  Here a  dichotomy  emerges.  Complex notions  of similar(cid:173)\nity  appear necessary for  cognitive models  and applications,  while  simple \nnotions of similarity  form a tractable basis for current computational ap(cid:173)\nproaches  to classification.  We  explore  the  nature of this dichotomy  and \nwhy  it  calls  for  new  approaches  to  well-studied  problems  in  learning. \nWe  begin  this  process  by  demonstrating  new  computational  methods \nfor  supervised  learning  that  can  handle  complex  notions  of similarity. \n(1)  We  discuss  how  to  implement  parametric  met.hods  that  represent  a \nclass  by  its  mean  when  using  non-metric  similarity  functions;  and  (2) \nWe review  non-parametric  methods  that we  have developed  using  near(cid:173)\nest  neighbor  classification  in  non-metric  spaces.  Point  (2) ,  and  some of \nthe  background  of our  work have been described  in  more detail  in  [8]. \n\n1  Supervised Learning and  Non-Metric Distances \n\nHow  can  one  represent  one 's  knowledge  of previously  encountered  objects  in  order \nto  classify new  objects?  We study this question within the framework of supel vised \nlearning:  it is assumed that one is given a number of training objects, each labeled as \nbelonging to a category; one wishes to use this experience to label new  test instances \nof objects.  This  problem emerges  both in  the modeling of cognitive  processes  and \nin  many  practical  applications.  For  example,  one  might  want  to  identify  risky \napplicants for  credit  based  on  past  experience  with clients  who  have  proven  to  be \ngood or  bad credit  risks.  Our  work  is  motivated by computer vision  applications. \n\nMost current computational approaches to supervised  learning suppose that objects \ncan  be  thought  of as  vectors  of numbers,  or  equivalently  as  points  lying  in  an  n(cid:173)\ndimensional space.  They further suppose that the similarity between objects can be \ndetermined from  the Euclidean distance  between  these  vectors, or from some other \nsimple metric.  This classic notion of similarity as Euclidean or metric distance leads \n\n\fClassification in Non-Metric Spaces \n\n839 \n\nto considerable mathematical and computational simplification . \n\nHowever,  work  in  cognitive  psychology  has  challenged  such  simple notions of sim(cid:173)\nilarity  as  models  of human  judgment,  while  applications  frequently  employ  non(cid:173)\nEuclidean distances  to measure object similarity.  We consider  the need for  similar(cid:173)\nity measures that are not only non-Euclidean , but that are non-metric.  We focus on \nproposed similarities that violate one requirement of a metric distance, the triangle \ninequality.  This  states  that  if we  denote  the  distance  between  objects  A  and  B \nby  d(A , B) , then :  VA , B , C  : d(A, B) + d(B, C)  ~ d(A , C) .  Distances  violating the \ntriangle inequality must also be non-Euclidean. \n\nData from  cognitive  psychology  has  demonstrated  that  similarity judgments  may \nnot  be  well  modeled  by  Euclidean  distances.  Tversky  [12]  has  demonstrated  in(cid:173)\nstances  in  which  similarity judgments may  violate  the  triangle inequality.  For  ex(cid:173)\nample, close  similarity between  Jamaica and  Cuba and  between  Cuba and  Russia \ndoes  not  imply close  similarity between  Jamaica and  Russia  (see  also  [10]) .  Non(cid:173)\nmetric  similarity measures  are  frequently  employed  for  practical  reasons,  too  (cf. \n[5]) .  In  part,  work  in  robust statistics  [7]  has shown  that methods that will survive \nthe  presence  of outliers, which  are extraneous  pieces  of information or  information \ncontaining extreme errors, must employ non-Euclidean distances that in fact violate \nthe  triangle inequality ; related  insights  have spurred  the  widespread  use  of robust \nmethods in computer vision  (reviewed  in  [5]  and  [9]). \n\nWe are interested in handling a wide range of non-metric distance functions,  includ(cid:173)\ning  those  that are so  complex that they  must  be  treated  as  a  black  box .  However, \nto be concrete,  we  will focus  here  on  two  simple examples of such  distances: \n\nmedian distance:  This  distance  assumes  that  objects  are  representable  as  a  set \nof features  whose  individual  differences  can  be  measured,  so  that  the  difference \nbetween  two  objects  is  representable  as  a  vector:  J = (d1 , d2 ,  .. . dn ).  The  median \ndistance  between  the two objects  is just the median value in  this  vector.  Similarly, \none can define a k-median distance by choosing the k'th lowest element in this list.  k(cid:173)\nmedian distances are often used in applications (cf.  [9]) , because they are unaffected \nby the exact values of the most extreme differences  between  the objects .  Only these \nfeatures  that  are  most  similar  determine  its  value.  The  k-median  distance  can \nviolate  the triangle inequality  to  an  arbitrary degree  (i.e. , there  are  no constraints \non  the  pairwise distances  between  three  points) . \nrobust  non-metric  LP  distances:  Given  a  difference  vector  J,  an  LP  distance \nhas  the form: \n\n(1) \n\nand is  non-metric for  p < 1. \nFigure  1  illustrates  why  these  distances  present  significant  new  challenges  in  su(cid:173)\npervised  learning.  Suppose  that given  some datapoints  (two in  Fig.  1) , we  wish  to \nclassify  each  new  point  as coming from  the  same category  as  its  nearest  neighbor. \nThen  we  need  to determine the Voronoi  diagram generated  by our data:  a  division \nof the  plane  into  regions  in  which  the  points  all  have  the  same  nearest  neighbor. \nFig.  1 shows  how  the Voronoi diagram changes  with  the function  used  to compute \nthe distance between datapoints; the non-metric diagrams (rightmost three pictures \nin  Fig.  1)  are more complex and  more likely  to make non-intuitive predictions.  In \nfact , very  little is  known about  the computation of non-metric  Voronoi diagrams. \n\nWe  now  describe  new  parametric methods for  supervised  learning with  non-metric \n\n\f840 \n\nD. Weins hall,  D.  W Jacobs and Y.  Gdalyahu \n\nFigure  1:  The Voronoi  diagram  for  two  points  using,  from  left  to right,  p-distances  with \np  =  2  (Euclidean),  p  =  1  (  Manhattan,  which  is  still  metric),  the  non-metric  distances \narising  from  p  =  0.5,  p  =  0.2,  and  the min  (I-median)  distance.  The min distance  in  2-D \nillustrates  the behavior of the other median distances  in  higher  dimensions.  The region of \nthe  plane  closer  to one point  is  shown  in  black, and closer  to the other in  white. \n\ndistances,  and review  non-parametric methods that we  described  in  [8]. \n\n2  Parametric methods:  what  should  replace the  mean \n\nParametric  methods  typically  represent  objects  as  vectors  in  a  high-dimensional \nspace,  and  represent  classes  and  the  boundaries  between  them  in  this  space  us(cid:173)\ning  geometric  constructions  or  probability  distributions  with  a  limited  number  of \nparameters.  One  can  attempt  to  extend  these  techniques  to  specific  non-metric \ndistances,  such  as  the  median  distance ,  or  non-metric  LP  distances.  We  discuss \nthe  example  of the  mean  of a  class  below.  One  can  also  redefine  geometric  ob(cid:173)\njects such  as  linear separators,  for  specific  non-metric distances.  However,  existing \nalgorithms for  finding  such  objects  in  Euclidean  spaces  will  no  longer  be  directly \nsuitable,  nor  will  theoretical  results  about  such  representations  hold.  Many  prob(cid:173)\nlems  are  therefore  open  in  determining  how  to  best  apply  parametric supervised \nlearning techniques  to specific  non-metric distances. \n\n1 \n\nWe analyze k-means clustering where each class is  represented  by its average mem(cid:173)\nber;  new elements are then classified according to which of these  prototypical exam(cid:173)\nples  is  nearest .  In  Euclidean space,  the mean is  the  point q whose  sum of squared \ndistances  to all  the class members {qdr=l  - (2:~1 d(ij, qi)2)2  - is  minimized. \nSuppose  now  that  our  data  come  from  a  vector  space  where  the  correct  distance \nis  the  LP  distance  from  (1).  Using  the  natural  extension  of the  above  definition, \nwe  should  represent  each  class  by  the  point  ij  whose  sum  of distances  to  all  the \nclass  members - (2:~=1 d(ij, qi)P) p  - is  minimal.  It is  now  possible  to  show  (proof \nis  omitted)  that for  p  <  1  (the  non-metric cases),  the  exact  value  of every  feature \nof the representative  point ij must have already appeared in  at least one element in \nthe  class.  Moreover,  the  value of these features  can be determined separately  with \ncomplexity O(n 2 ),  and  total  complexity of O(dn 2 )  given  d features .  ij  is  therefore \ndetermined  by  a  mixture  of up  to  d  exemplars,  where  d is  the  dimension  of the \nvector  space.  Thus there  are efficient  algorithms for  finding  the  \"mean\"  element of \na  class,  even  using certain non-metric distances. \n\n1 \n\nWe  will  illustrate  these  results  with  a  concrete  example using  the  corel  database, \na  commercial database of images pre-labeled  by categories  (such  as  \"lions\"), where \nnon-metric distance functions have  proven  effective  in  determining the similarity of \nimages [1] .  The corel database is very large, making the use of prototypes desirable. \n\nWe  represent  each  image  using  a  vector  of  11  numbers  describing  general  image \nproperties,  such  as color histograms, as described  in  [1] .  We consider  the Euclidean \n\n\fClassification in  Non-Metric Spaces \n\n841 \n\nand  L0 5  distances,  and  their  corresponding  prototypes:  the  mean  and  the  LO.5_ \nprototype computed according  to the  result  above.  Given  the first  45  classes, each \ncontaining 100 images , we found their corresponding prototypes; we  then computed \nthe percentage of images in each class that are closest  to their own  prototype, using \neither the Euclidean or the L 0.5  distance and one of the two prototypes.  The results \nare the following: \n\nmean  d existing features \n\ndistance \n\nEuclidean distance \n\n25% \n20  0 \n\nIn  the first  column , the  prototype  is  computed using  the  Euclidean  mean.  In  the \nsecond  column  the  prototype  is  computed  using  an  LO  5  distance.  In  each  row ,  a \ndifferent  function  is  used  to  compute  the  distance  from  each  item  to  the  cluster \nprototype.  Best  results  are  indeed  obtained with  the non-metric  L05  distance  and \nthe  correct  prototype for  this  particular  distance.  While  performance  in  absolute \nterms depends on how  well  this data clusters  using distances derived from  a simple \nfeature  vector,  relative  performance  of different  methods  reveals  the  advantage  of \nusing  a  prototype computed with a  non-metric distance. \n\nAnother  important distance  function  is  the  generalized  Hamming distance:  given \ntwo  vectors  of features,  their  distance  is  the  number  of features  which  are  differ(cid:173)\nent  in  the  two  vectors.  This  distance  was  assumed  in  psychophysical  experiments \nwhich  used  artificial objects  (Fribbles)  to investigate human categorization and ob(cid:173)\nject  recognition  [13].  In  agreement  with  experimental  results ,  the  prototype  if  for \nthis distance  computed  according  to the  definition  above  is  the  vector  of  \"modal\" \nfeatures - the most common feature value computed independently  at each feature. \n\n3  Non-Parametric Methods:  Nearest  Neighbors \n\nNon-parametric  classification  methods  typically  represent  a  class  directly  by  its \nexemplars.  Specifically, nearest-neighbor techniques  classify new objects using only \ntheir  distance  to labeled exemplars.  Such  methods can be  applied  using  any  non(cid:173)\nmetric distance function ,  treating  the  function  as  a  black-box.  However,  nearest(cid:173)\nneighbor  techniques  must  also  be  modified  to  apply  well  to  non-metric distances. \nThe insights we  gain below from doing this can form the basis of more efficient  and \neffective computer algorithms, and of cognitive models for  which examples of a class \nare worth  remembering.  This section  summarizes work described  in  [8]. \n\nCurrent  efficient  algorithms for  finding  the  nearest  neighbor  of a  class  work  only \nfor  metric distances  [3].  The alternative of a  brute-force  approach,  in which  a  new \nobject  is  explicitly  compared  to  every  previously  seen  object ,  is  desirable  neither \ncomputationally  nor  as  a  cognitive  model.  A  natural  approach  to  handling  this \nproblem is  to represent  each  class  by  a  subset  of its labeled  examples.  Such  meth(cid:173)\nods  are  called  condensing  algorithms.  Below  we  develop  condensing  methods  for \nselecting  a  subset  of the training set  which  minimizes errors  in the classification  of \nnew datapoints, taking into account  the non-metric nature of the distance. \nIn  designing  a  condensing  method , one  needs  to  answer  the  question  when  is  one \nobject  a  good  substitute for  another?  Earlier  methods  (e.g.,  [6,  2])  make use  of the \nfact  that the triangle inequality guarantees that when two points are similar to each \nother,  their  pattern  of similarities to  other  points  are  not  very  different .  Thus,  in \na  metric  space,  there  is  no  reason  to  store  two  similar datapoints,  one can  easily \nsubstitute for  the other.  Things are different  in  non-metric spaces. \n\n\f842 \n\nD.  Weinshall,  D.  W Jacobs and Y.  Gdalyahu \n\na \n\nFigure 2:  a)  Two clusters of labeled  points  (left)  and  their Voronoi  diagram  (right)  com(cid:173)\nputed using  the I-median (min) distance.  Cluster P  consists of four points  (black squares) \nall  close together both according  to the median distance and the Euclidean  distance.  Clus(cid:173)\nter  Q consists  of five  points  (black  crosses)  all  having  the  same  x  coordinate,  and  so  all \nare separated by zero distance using  the median  (but not Euclidean)  distance.  We  wish to \nselect  a  subset  of points  to represent  each  class,  while  changing  this  Voronoi  diagram  as \nlittle  as  possible.  b)  All  points  in  class  Q  have  zero  distance  to each other,  using  the min \ndistance.  So  distance provides no  clue  as  to  which  are interchangeable.  However,  the top \npoints  (ql, q2)  have  distances  to the points  in  class  P  that are highly  correlated with each \nother, and poorly  correlated with the bottom points (q3, q4, qs).  Without  using correlation \nas  a  clue,  we  might  represent  Q  with  two points  from  the  bottom  (which  are nearer  the \nboundary  with  P,  a  factor  preferred  in  existing  approaches).  This  changes  the  Voronoi \ndiagram drastically,  as shown on the left.  Using correlation as a clue,  we select points from \nthe top and  bottom,  changing  the Voronoi  diagram  much less,  as  shown on  the right. \n\nSpecifically,  what we  really need  to know is when  two objects will  have similar dis(cid:173)\ntances to other objects,  yet unseen.  We estimate this quantity  using the correlation \nbetween  two  vectors:  the  vector  of distances  from  one  datapoint  to  all  the  other \ntraining data,  and  the  vector  of distances from  the second  datapoint  to all  the  re(cid:173)\nmaining training datal.  It can be shown (proof is omitted) that in a  Euclidean space \nthe similarity between  two points is the best measure of how well one can substitute \nthe other,  whereas in  a  non-metric space  the aforementioned vector correlation is a \nsubstantially better measure.  Fig.  2  illustrates this result . \n\nWe now draw on these insights to produce concrete methods for representing classes \nin  non-metric  spaces,  for  nearest  neighbor  classification.  We  compare  three  algo(cid:173)\nrithms.  The  first  two  algorithms,  random  selection  (cf. \n[6])  and  boundary \ndetection (e.g.,  [11]),  represent  old condensing ideas:  in the first  we pick a  random \nselection  of class representatives,  in  the second  we  use  points close  to class  bound(cid:173)\naries as representatives.  The last  algorithm uses  new  ideas:  correlation selection \nincludes  in  the  representative  set  points  which  are  least  correlated  with  the other \nclass  members  and  representatives.  To  be  fair  in  our  comparison,  all  algorithms \nwere  constrained  to select  the same number of representative  points for  each class. \n\nDuring the simulation, each  of 1000  test datapoints was classified  based  on:  (1)  all \nthe  data,  (2)  the  representatives  computed  by  each  of the  three  algorithms.  For \neach  algorithm, the test  is successful  if the two methods  (classification  based on  all \nthe data and  based on  the chosen  representatives)  give  the same results.  Fig.  3a-c \nsummarizes representative  results of our simulations.  See  [8]  for  details. \n\nIGiven  two  datapoints  X, Y  and  x, y  ERn, where x  is  the vector of distances  from  X \nto all  the other  training  points  and y  is  the corresponding  vector  for  Y,  we  measure  the \ncorrelation between the datapoints using the statistical correlation coefficient between x, y: \ncorr(X, Y) = corr(x, y) = ~. Y-I-'y,  where JJx,  JJy  denote the mean of x, y  respectively, \nand  frx , fry  denote  the standard deviation  of x, y  respectively. \n\nCTy \n\nCTx \n\n\fClassification in Non-Metric Spaces \n\n843 \n\n[p. ... \n\n.. \n\n..)~ \n, \n-- / \n\n~ \n\nf----\n\ncorrelation  D\u00b7 \nboundary  +-\n\nrandom  ~ \n\nmedian  L\u00b70.2  L\u00b70.5 Euclidean \n\nb) \n\nti \n~ \n... \nc: \nOJ \n~ \n\n0. \n\nti \n~ \n8 \nc: \n~ \nOJ \n0. \n\n100 \n\n90 \n\n80 \n\n70 \n\n60 \n\n50 \n\n40 \n\n100 \n\n90 \n\n80 \n\n70 \n\n60 \n\n50 \n\n40 \n\n~ '. \n\n.''1....-.... \n\n...-\" \n\nti \n\n\u00a7 \nc: \n~ \nOJ \n0. \n\ncorrelation  D\u00b7 \nboundary  +-\n\nrandom  ~ \n\nmedian  L\u00b70.2  L\u00b70.5 Euclidean \n\na) \n\n\u00b7\u00b7\u00b7\u00b7-m .... \u00b7-m- .. \u00b7;;i \n\n, , \n\nt  t//(~ \nf/\"!' \n\ncorrelation  D\u00b7 \nboundary  +-\n\nrandom  ~ \n\nmedian  L\u00b70.2  L\u00b70.5 Euclidean \n\nC) \n\nti \n~ \n... \n0 \nc: \n~ \nOJ \n0. \n\n100 \n\n90 \n\n80 \n\n70 \n\n60 \n\n50 \n\n40 \n\n100 \n\n60 \n\n50 \n\n40 \n\ns .. .... .. \u00b7 E!j \n\n80 \n\n90  14 \n\n70 \n\ncorrelation  D\u00b7 \nboundary  +-\n\nrandom  ~ \n\n5 reps \n\n7 reps \n\nd) \n\nFigure 3: Results:  values of percent correct scores, as well as error bars giving the standard \ndeviation  calculated  over  20  repetitions  of each  test  block  when  appropriate.  Each graph \ncontains 3 plots , giving  the percent correct score for each of the three algorithms described \nabove:  random  (selection),  boundary  (detection),  and  (selection  based  on)  correlation. \n(a-c)  Simulation  results:  data is  chosen from  R2 5 .  30 clusters were randomly  chosen, each \nwith  30  datapoints.  The distribution  of points in each class  was:  (a) normal;  (b)  normal, \nwhere in half the datapoints  one random coordinate  was  modified  (thus  the points cluster \naround  a  prototype,  but  many  class  members  vary  widely  in  one  random  dimension);  (c) \nunion of 2 concentric normal distributions,  one spherical  and one elongated elliptical  (thus \nthe  points  cluster  around  a  prototype,  but  may  vary  significantly  in  a  few  non-defining \ndimensions) .  Each  plot  gives  4  values,  for  each  of  the  different  distance  functions  used \nhere:  median,  Lo.2 ,  Lo.5  and  L2 .  (d)  Real  data:  the number of representatives  chosen  by \nthe  algorithm  was limited  to 5  (first  column)  and 7  (second column) . \n\nTo  test  our method with  real  images,  we  used  the  local  curve  matching  algorithm \ndescribed in [4].  This non-metric curve matching algorithm was specifically designed \nto  compare curves  which  may  be  quite different,  and  return  the  distance  between \nthem .  The training and test  data are shown in Fig. 4 .  Results  are given in  Fig.  3d. \n\nThe  simulations and  the  real  data demonstrate a  significant  advantage  to  our  new \nmethod .  Almost as  important , in  metric spaces  (4th column in  Fig.  3a-c)  or when \nthe classes  lack any  \"interesting\"  structure  (Fig . 3a) , our method is not  worse than \nexisting methods.  Thus it should  be used to guarantee good  performance when the \nnature of the data and  the distance function  is  not  known  a  priori. \n\nReferences \n\n[1]  Cox,  I. ,  Miller,  M.,  Omohundro ,  S. ,  and  Yianilos,  P.,  1996,  \"PicHunter:  Bayesian \n\nRelevance  Feedback  for  Image  Retrieval,\"  Proc.  of I CPR,  C:361- 369. \n\n\f844 \n\nD.  Weinshall,  D.  W Jacobs and Y.  Gdalyahu \n\n,.............  ~~ \n~ l)1V  ~ ~ o=C0  ~ \n~ \n\\-.  ~~ \nW>~~ OJcCflp \n~~~ c/C;'J  ~ cfY  1~) \n1yl1  P  c!fl  P  ~);J N \n\na) \n\nb) \n\nc) \n\nd) \n\nFigure  4:  Real  data used  to  test the three algorithms,  incluillng  2 classes  with 30 images \neach:  a)  12  examples  from  the  first  class  of  30  cow  contours,  obtained  from  illfferent \nviewpoints  of  the  same  cow.  b)  12  examples  from  the  second  class  of 30  car  contours, \nobtained from  different  viewpoints of 2 similar cars.  c) 12 examples from  the set of 30 test \ncow contours,  obtained from illfferent  viewpoints  of the same cow with possibly  adilltional \nocclusion.  d)  2 examples  of the real images  from  which  the contours in  a)  are obtained. \n\n[2]  Dasarathy,  B.,  1994,  \"Minimal  Consistent Set (MCS) Identification for Optimal  Near(cid:173)\nest  Neighbor  Decision  Systems  Design,\"  IEEE  Trans.  on  Systems,  Man  and  Cyber(cid:173)\nnetics,24(3):511-517. \n\n[3]  Friedman,  J.,  Bently,  J .,  Finkel,  R.,  1977,  \"An  Algorithm  for  Finillng  Best  Matches \n\nin Logarithmic  Expected Time,\"  ACM Trans.  on  Math.  Software,  3:3  209-226. \n\n[4]  Gdalyahu,  Y. and D.  Weinshall,  1997,  \"Local Curve Matching for Object Recognition \n\nwithout  Prior Knowledge\",  Proc .:  DARPA  Image  Understanding  Workshop,  1997. \n[5]  Haralick,  R.  and  L.  Shapiro,  1993,  Computer  and  Robot  Vision,  Vol.  2,  Addison(cid:173)\n\nWesley  Publishing. \n\n[6]  Hart, P., 1968,  \"The Condensed Nearest Neighbor Rule,\"  IEEE Trans.  on Information \n\nTheory,  14(3):515- 516. \n\n[7]  Hubbr,  P.,  1981,  Robust Statistics, John Wiley  and Sons. \n[8]  Jacobs,  D.,  Weinshall,  D.,  and  Gdalyahu,  Y.,  1998,  \"Condensing  Image  Databases \nwhen Retrieval is  based on Non-Metric Distances,\"  Int.  Conf.  on Computer  vis.:596-\n60l. \n\n[9]  Meer,  P., D. Mintz,  D. Kim and A. Rosenfeld,  1991,  \"Robust  Regression  Methods for \n\nComputer Vision:  A  Review,\"  Int.  J.  of Compo  Vis.  6(1):59-70. \n\n[10]  Rosch,  E .,  1975,  \"Cognitive  Reference  Points,\"  Cognitive Psychology, 7:532-547. \n[11]  Tomek,  1.,  1976,  \"Two  moillfications  of CNN,\"  IEEE  Trans.  Syst. ,  Man,  Cyber.\" \n\nSMC-6{1l):769-772. \n\n[12]  Tversky,  A.,  1977,  \"Features of Similarity,\"  Psychological Review, 84(4):327- 352. \n[13]  Williams,  P.,  \"Prototypes,  Exemplars,  and Object Recognition\",  submitted. \n\n\fPART VIII \n\nApPLICATIONS \n\n\f\f", "award": [], "sourceid": 1581, "authors": [{"given_name": "Daphna", "family_name": "Weinshall", "institution": null}, {"given_name": "David", "family_name": "Jacobs", "institution": null}, {"given_name": "Yoram", "family_name": "Gdalyahu", "institution": null}]}