Low dimensional representations and multidimensional scaling mds sec 10 14
This presentation is the property of its rightful owner.
Sponsored Links
1 / 6

Low Dimensional Representations And Multidimensional Scaling (MDS ) (Sec 10.14) PowerPoint PPT Presentation


  • 78 Views
  • Uploaded on
  • Presentation posted in: General

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

Download Presentation

Low Dimensional Representations And Multidimensional Scaling (MDS ) (Sec 10.14)

An Image/Link below is provided (as is) to download presentation

Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author.While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server.


- - - - - - - - - - - - - - - - - - - - - - - - - - E N D - - - - - - - - - - - - - - - - - - - - - - - - - -

Presentation Transcript


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


Distances between g iven p oints is meaningful

Distances Between Given Points is 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


  • Login