CS 536 : Support Vector Machine Problems

1) Suppose you had a data set in two dimensions that satisfied the following: the positive class all lay within a

certain radius of a point, the negative class all lay outside that radius.

– Show that under the feature map φ(x1, x2) = (1, x1, x2, x1x2, x2

1

, x2

2

) (or equivalently, with the kernel

K(x, y) = (1 + x.y)

2

), a linear separator can always be found in this embedded space, regardless of radius

and where the data is centered.

– In fact show that if there is an ellipsoidal separator, regardless of center, width, orientation (and dimension!), a separator can be found in the quadratic feature space using this kernel.

2) As an extension of the previous problem, suppose that the two dimensional data set satisfied the following: the

positive class lay within one of two (disjoint) ellipsoidal regions, and the negative class was everywhere else.

Argue that the kernel K(x, y) = (1 + x.y)

4 will recover a separator.

3) Suppose that the two dimensional data set is distributed like the following: the positive class lays in a circle

centered at some point, the negative class lies in a circular band surrounding it of some radius, and then

additional positive points lie outside that radius. Argue that the kernel K(x, y) = (1 + x.y)

4 will recover a

separator.

4) Consider the XOR data (located at (±1, ±1)). Express the dual SVM problem and show that a separator can

be found using

– K(x, y) = (1 + x.y)

2

– K(x, y) = exp(−||x − y||2

).

For each, determine the regions of (x1, x2) space where points will be classified as positive or negative. Given

that each produces a distinct separator, how might you decide which of the two was preferred?

1

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# CS 536 : Support Vector Machine Problems

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