1. Let w = c+ −c− and let b = 12 (                c−           2 −          c+           2). Show that h(x) = sign(_w,ψ(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 S of examples in Rn × [k] for which there exist vectors μ1, . . .,μk such that every example (x, y) ∈ S falls within a ball centered at μy whose radius is r ≥ 1. Assume also that for every i _= j ,          μi −μj     ≥ 4r . Consider concatenating each instance by the constant 1 and then applying the multivector construction, namely, _(x, y)= [ :0, .;.. ,0= ∈R(y−1)(n+1) , :x1, . .;., xn ,1= ∈Rn+1 , :0, .;.. ,0= ∈R(k−y)(n+1) ].

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

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