Show how to express h(x) on the basis of the kernel function, and without accessing individual entries of ψ(x) or w.

1. Let w = c+ −c− and let = 12 (                c−           2 −          c+           2). Show that h(x) = sign(_w(x)_+b).

2. Show how to express h(x) on the basis of the kernel function, and without accessing individual entries of ψ(x) or w.

3. Consider a set of examples in R× [k] for which there exist vectors μ1, . . .,μsuch that every example (xy) ∈ falls within a ball centered at μy whose radius is ≥ 1. Assume also that for every _= ,          μμj     ≥ 4. Consider concatenating each instance by the constant 1 and then applying the multivector construction, namely, _(xy)= [ :0, .;.. ,0= ∈R(y−1)(n+1) :x1, . .;.xn ,1= ∈Rn+1 :0, .;.. ,0= ∈R(ky)(n+1) ].

Show that there exists a vector w ∈ Rk(n+1) such that    (w(xy)) = 0 for every (xy) ∈ S.

Hint: Observe that for every example (xy) ∈ we can write x = μ+v for some                v              ≤r. Now, take w = [w1, . . .,w], where w= [μi −               μi            2/2].

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