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

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

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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 (MDSGiven Points is Meaningful

Criterion Functions (MDS

- 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 (MDS

- Use gradient-descent procedure to find an optimal configuration y1, …, yn

Example (MDS

20 iterations with Jef

Nonmetric (MDS 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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