Topographic processing of relational data

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Topographic processing of relational data
Barbara Hammer, TU Clausthal, Germany Alexander
Hasenfuss, TU Clausthal, Germany Fabrice Rossi,
INRIA, France Marc Strickert, IPK Gatersleben,
Germany
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• Topographic processing of euclidean data
• SOM
• NG
• Topographic processing of relational data
• Median clustering
• Relational clustering
• Supervision
• Experiments
• Proteins
• Chromosome images
• Macroarrays

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Topographic processing of euclidean data
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SOM and NG

• Prototype based clustering
• prototypes element of data space
• clustering by means of receptive fields
• euclidean distance

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SOM
Self-organizing map initialize wi adapt wi
exp(-rk(xj,wi)/s)(xj-wi)
almost optimizes ESOM ?ij?j(i)?kexp(-nd(i,k
)/s) (xj-wk)2
direct visualization in euclidean or hyperbolic
space
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Neural Gas
almost optimizes ESOM ?ij?j(i)?kexp(-nd(i,k
)/s) (xj-wk)2
kij
Batch-SOM repeat
optimize kij given fixed w
optimize w given fixed kij i.e.
repeat kij 1 iff wi
winner for xj , I(xi) winner
wi ?j exp(-nd(I(xj),i)/s) xj / ?j
exp((-nd(I(xj),i)/s) converges, (quadratic)
Newton scheme
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Neural Gas
Neural Gas initialize wi adapt wi
exp(-rk(xj,wi)/s)(xj-wi)
optimizes ENG ?ijexp(-rk(xj,wi)/s) (xj-wi)2
neighborhood graph induced by Delaunay
triangulation visualization by means of MDS or
similar
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Neural Gas
optimizes ENG ?ijexp(-rk(xj,wi)/s) (xj-wi)2
kij
Batch-NG repeat optimize
kij given fixed w optimize
w given fixed kij i.e. repeat
kij rank of prototype wi given
xj wi ?j exp(-kij/s) xj
/ ?j exp(-kij/s) converges, (quadratic) Newton
scheme
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Topographic processing of noneuclidean data
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left angle
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median clustering restrict prototypes to data
positions relational clustering substitute
distance from prototype
dij
method to represent wi and compute distances

• dissimilarity matrix D or data
• no (explicit) euclidean embedding

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Median clustering

• Median clustering
• no assumptions on D
• prototypes restricted to data points ? only
  discrete values
• consecutive optimization of kij and wi as
  beforehand

Median-NG kij rank of
prototype wi given xj wi xk
with ?ij exp(-kij/s) (xj-xk)2 minimum
avoid identical prototypes!
converges for every D
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Relational clustering

• Relational clustering
• d(xi,xj) f(xi)-f(xj)2 is euclidean, but
  embedding unknown

optimum prototypes fulfill wi ?l ail xl where
?ail 1 normalized ranks ? xj-wi2 (D ai)j
½ ait D ai ? dual cost function ENG ?i?ll
exp(-rk(xl,wi)/s) exp(-rk(xl,wi)/s) d(xl,xl)2 /
4 ?lexp(-rk(xl,wi)/s)
Relational-NG xj-wi2 (D ai)j ½ ait D
ai kij rank of prototype wi
given xj aij exp(-kij/s),
normalize
only implicit prototypes represented by aij ?
continuous adaptation converges for every
symmetric nonsingular D
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Supervision

• integrate additional label information for data
  yi
• enrich prototypes by labels Yj
• substitute distance
• xi-wj2 ? ß xi-wj2 (1-ß) yi-Yj2
• solve as beforehand

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Experiments
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Proteins

• Protein classifiation
• 226 points, 5 classes (HA, HB, MY, GG/GP, other)
• alignment measures evolutionary distance of
  globin proteins
• 29 neurons, 150 epochs,
  repeated
  cross-validation,
  
  mixing
  parameter 0.5

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Proteins

• Visualization

NG MDS
HSOM
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Chromosomes

• Kopenhagen Chromosome database
• 4200 points, 22 classes, alignment distance
• difference of thickness profiles of grey images
• 85 neurons, 100 epochs,
  repeated
  cross-validation,
  mixing
  parameter 0.9

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Macroarray data

• Macroarray data
• gene expressions at 14 time points after
  flowering
• 4824 selected genes
• 85 epochs, 150 neurons, no supervision
• Pearson correlation

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