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

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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)

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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

### 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

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

### Example

20 iterations with Jef

### 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