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Low Dimensional Representations And Multidimensional Scaling (MDS ) (Sec 10.14). Given n points (objects) x 1 , …, x n . No class labels Suppose only the similarities between the n objects are provided

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low dimensional representations and multidimensional scaling mds sec 10 14
Low Dimensional Representations And Multidimensional Scaling (MDS) (Sec 10.14)
  • Given n points (objects) x1, …, xn . No class labels
  • Suppose only the similarities between the n objects are provided
  • Goal is to represent these n objects in some low dimensional space in such a way that the distances between points in that space corresponds to the dissimilarities in the original space
  • If an accurate representation can be found in 2 or 3 dimensions than we can visualize the structure of the data
  • Find a configuration of points y1, …, ynfor which the n(n-1) distances dij are as close as possible to the original similarities; this is called Multidimensional scaling
  • Two cases
    • Meaningful to talk about the distances between given n points
    • Only rank order among similarities are meaningful
criterion functions
Criterion Functions
  • Sum of squared error functions
  • Since they only involve distances between points, they are invariant to rigid body motions of the configuration
  • Criterion functions have been normalized so their minimum values are invariant to dilations of the sample points
finding the optimum configuration
Finding the Optimum Configuration
  • Use gradient-descent procedure to find an optimal configuration y1, …, yn
example
Example

20 iterations with Jef

nonmetric multidimensional scaling
Nonmetric Multidimensional Scaling
  • Numerical values of dissimilarities are not as important as their rank order
  • Monotonicityconstraint: rank order of dij = rank order of ij
  • The degree to which dij satisfy the monotonicy constraint is measured by
  • Normalize to prevent it from being collapsed
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