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Assume you have two datasets of ( x , y ) : D 1 = { ( 1 , + 1 ) , ( 0

Assume you have two datasets of (x, y):
D1={(1,+1),(0,1),(1,+1)}.
D2={(1,1),(0,+1),(1,1)}.
If you use feature \phi (x)= x, neither dataset can be linearly separated. You need to define a
two-dimensional feature \phi (x) to fix this such that:
A weight vector w1 can classify D1 perfectly (i.e., w1\phi (x)>0 if x has label +1 and
w1\phi (x)<0 if x has label 1); and
A weight vector w2 can classify D2 perfectly;
Note the two datasets share the same features \phi (x) but the weight vectors can be different. \phi (x)= x
normally can be re-written as \phi (x)=[1, x], which is regarded as one-dimensional feature.

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