Classification in Non-Metric Spaces
Authors: David W. Jacobs, Yoram Gdalyahu Daphna Weinshall
Published in: Advances in Neural Information Processing Systems 11  (NIPS 1998) (Main Conference Track), 1998

Abstract: A  key question in vision  is  how to represent our knowledge  of previously  encountered objects to classify new ones.  The answer depends on how we  determine  the similarity  of two  objects.  Similarity  tells  us how  relevant  each previously seen object is in determining the category to which a new  object belongs.  Here a  dichotomy  emerges.  Complex notions  of similar(cid:173) ity  appear necessary for  cognitive models  and applications,  while  simple  notions of similarity  form a tractable basis for current computational ap(cid:173) proaches  to classification.  We  explore  the  nature of this dichotomy  and  why  it  calls  for  new  approaches  to  well-studied  problems  in  learning.  We  begin  this  process  by  demonstrating  new  computational  methods  for  supervised  learning  that  can  handle  complex  notions  of similarity.  (1)  We  discuss  how  to  implement  parametric  met.hods  that  represent  a  class  by  its  mean  when  using  non-metric  similarity  functions;  and  (2)  We review  non-parametric  methods  that we  have developed  using  near(cid:173) est  neighbor  classification  in  non-metric  spaces.  Point  (2) ,  and  some of  the  background  of our  work have been described  in  more detail  in  [8]. 
1  Supervised Learning and  Non-Metric Distances 
How  can  one  represent  one 's  knowledge  of previously  encountered  objects  in  order  to  classify new  objects?  We study this question within the framework of supel vised  learning:  it is assumed that one is given a number of training objects, each labeled as  belonging to a category; one wishes to use this experience to label new  test instances  of objects.  This  problem emerges  both in  the modeling of cognitive  processes  and  in  many  practical  applications.  For  example,  one  might  want  to  identify  risky  applicants for  credit  based  on  past  experience  with clients  who  have  proven  to  be  good or  bad credit  risks.  Our  work  is  motivated by computer vision  applications. 
Most current computational approaches to supervised  learning suppose that objects  can  be  thought  of as  vectors  of numbers,  or  equivalently  as  points  lying  in  an  n(cid:173) dimensional space.  They further suppose that the similarity between objects can be  determined from  the Euclidean distance  between  these  vectors, or from some other  simple metric.  This classic notion of similarity as Euclidean or metric distance leads 
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to considerable mathematical and computational simplification . 
However,  work  in  cognitive  psychology  has  challenged  such  simple notions of sim(cid:173) ilarity  as  models  of human  judgment,  while  applications  frequently  employ  non(cid:173) Euclidean distances  to measure object similarity.  We consider  the need for  similar(cid:173) ity measures that are not only non-Euclidean , but that are non-metric.  We focus on  proposed similarities that violate one requirement of a metric distance, the triangle  inequality.  This  states  that  if we  denote  the  distance  between  objects  A  and  B  by  d(A , B) , then :  VA , B , C  : d(A, B) + d(B, C)  ~ d(A , C) .  Distances  violating the  triangle inequality must also be non-Euclidean. 
Data from  cognitive  psychology  has  demonstrated  that  similarity judgments  may  not  be  well  modeled  by  Euclidean  distances.  Tversky  [12]  has  demonstrated  in(cid:173) stances  in  which  similarity judgments may  violate  the  triangle inequality.  For  ex(cid:173) ample, close  similarity between  Jamaica and  Cuba and  between  Cuba and  Russia  does  not  imply close  similarity between  Jamaica and  Russia  (see  also  [10]) .  Non(cid:173) metric  similarity measures  are  frequently  employed  for  practical  reasons,  too  (cf.  [5]) .  In  part,  work  in  robust statistics  [7]  has shown  that methods that will survive  the  presence  of outliers, which  are extraneous  pieces  of information or  information  containing extreme errors, must employ non-Euclidean distances that in fact violate  the  triangle inequality ; related  insights  have spurred  the  widespread  use  of robust  methods in computer vision  (reviewed  in  [5]  and  [9]). 
We are interested in handling a wide range of non-metric distance functions,  includ(cid:173) ing  those  that are so  complex that they  must  be  treated  as  a  black  box .  However,  to be concrete,  we  will focus  here  on  two  simple examples of such  distances: 
median distance:  This  distance  assumes  that  objects  are  representable  as  a  set  of features  whose  individual  differences  can  be  measured,  so  that  the  difference  between  two  objects  is  representable  as  a  vector:  J = (d1 , d2 ,  .. . dn ).  The  median  distance  between  the two objects  is just the median value in  this  vector.  Similarly,  one can define a k-median distance by choosing the k'th lowest element in this list.  k(cid:173) median distances are often used in applications (cf.  [9]) , because they are unaffected  by the exact values of the most extreme differences  between  the objects .  Only these  features  that  are  most  similar  determine  its  value.  The  k-median  distance  can  violate  the triangle inequality  to  an  arbitrary degree  (i.e. , there  are  no constraints  on  the  pairwise distances  between  three  points) .  robust  non-metric  LP  distances:  Given  a  difference  vector  J,  an  LP  distance  has  the form: 
(1) 
and is  non-metric for  p < 1.  Figure  1  illustrates  why  these  distances  present  significant  new  challenges  in  su(cid:173) pervised  learning.  Suppose  that given  some datapoints  (two in  Fig.  1) , we  wish  to  classify  each  new  point  as coming from  the  same category  as  its  nearest  neighbor.  Then  we  need  to determine the Voronoi  diagram generated  by our data:  a  division  of the  plane  into  regions  in  which  the  points  all  have  the  same  nearest  neighbor.  Fig.  1 shows  how  the Voronoi diagram changes  with  the function  used  to compute  the distance between datapoints; the non-metric diagrams (rightmost three pictures  in  Fig.  1)  are more complex and  more likely  to make non-intuitive predictions.  In  fact , very  little is  known about  the computation of non-metric  Voronoi diagrams. 
We  now  describe  new  parametric methods for  supervised  learning with  non-metric 
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D. Weins hall,  D.  W Jacobs and Y.  Gdalyahu 
Figure  1:  The Voronoi  diagram  for  two  points  using,  from  left  to right,  p-distances  with  p  =  2  (Euclidean),  p  =  1  (  Manhattan,  which  is  still  metric),  the  non-metric  distances  arising  from  p  =  0.5,  p  =  0.2,  and  the min  (I-median)  distance.  The min distance  in  2-D  illustrates  the behavior of the other median distances  in  higher  dimensions.  The region of  the  plane  closer  to one point  is  shown  in  black, and closer  to the other in  white. 
distances,  and review  non-parametric methods that we  described  in  [8]. 
2  Parametric methods:  what  should  replace the  mean 
Parametric  methods  typically  represent  objects  as  vectors  in  a  high-dimensional  space,  and  represent  classes  and  the  boundaries  between  them  in  this  space  us(cid:173) ing  geometric  constructions  or  probability  distributions  with  a  limited  number  of  parameters.  One  can  attempt  to  extend  these  techniques  to  specific  non-metric  distances,  such  as  the  median  distance ,  or  non-metric  LP  distances.  We  discuss  the  example  of the  mean  of a  class  below.  One  can  also  redefine  geometric  ob(cid:173) jects such  as  linear separators,  for  specific  non-metric distances.  However,  existing  algorithms for  finding  such  objects  in  Euclidean  spaces  will  no  longer  be  directly  suitable,  nor  will  theoretical  results  about  such  representations  hold.  Many  prob(cid:173) lems  are  therefore  open  in  determining  how  to  best  apply  parametric supervised  learning techniques  to specific  non-metric distances. 
1 
We analyze k-means clustering where each class is  represented  by its average mem(cid:173) ber;  new elements are then classified according to which of these  prototypical exam(cid:173) ples  is  nearest .  In  Euclidean space,  the mean is  the  point q whose  sum of squared  distances  to all  the class members {qdr=l  - (2:~1 d(ij, qi)2)2  - is  minimized.  Suppose  now  that  our  data  come  from  a  vector  space  where  the  correct  distance  is  the  LP  distance  from  (1).  Using  the  natural  extension  of the  above  definition,  we  should  represent  each  class  by  the  point  ij  whose  sum  of distances  to  all  the  class  members - (2:~=1 d(ij, qi)P) p  - is  minimal.  It is  now  possible  to  show  (proof  is  omitted)  that for  p  <  1  (the  non-metric cases),  the  exact  value  of every  feature  of the representative  point ij must have already appeared in  at least one element in  the  class.  Moreover,  the  value of these features  can be determined separately  with  complexity O(n 2 ),  and  total  complexity of O(dn 2 )  given  d features .  ij  is  therefore  determined  by  a  mixture  of up  to  d  exemplars,  where  d is  the  dimension  of the  vector  space.  Thus there  are efficient  algorithms for  finding  the  "mean"  element of  a  class,  even  using certain non-metric distances. 
1 
We  will  illustrate  these  results  with  a  concrete  example using  the  corel  database,  a  commercial database of images pre-labeled  by categories  (such  as  "lions"), where  non-metric distance functions have  proven  effective  in  determining the similarity of  images [1] .  The corel database is very large, making the use of prototypes desirable. 
We  represent  each  image  using  a  vector  of  11  numbers  describing  general  image  properties,  such  as color histograms, as described  in  [1] .  We consider  the Euclidean 
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and  L0 5  distances,  and  their  corresponding  prototypes:  the  mean  and  the  LO.5_  prototype computed according  to the  result  above.  Given  the first  45  classes, each  containing 100 images , we found their corresponding prototypes; we  then computed  the percentage of images in each class that are closest  to their own  prototype, using  either the Euclidean or the L 0.5  distance and one of the two prototypes.  The results  are the following: 
mean  d existing features