Ex. 4.7

Ex. 4.7

Consider the criterion

D(β,β0)=i=1Nyi(xiTβ+β0),

a generalization of (4.41) in the textbook where we sum over all the observations. Consider minimizing D subject to β=1. Describe this criterion in words. Does it solve the optimal separating hyperplane problem?

Soln. 4.7

When β=1, βTxi+β0 is the signed distance of xi to the hyperplane βTx+β0=0. This does not solve the optimal separating hyperplane problem. Optimal separating hyperplane is actually solving a max-min problem such that each point satisfies the distance requirement, however minimizing D does not have such pointwise constraint.