Ex. 2.1

Ex. 2.1

Suppose each of K-classes has associated target tk, which is a vector of all zeros, except a one in the k-th position. Show that classifying to the largest of y^ amounts to choosing the closet target, minktky^, if the elements of y^ sum to one.

Soln. 2.1

We need to prove:

(1)argmaxky^k=argminktky^2

By definition of tk, we have

tky^2=(1y^k)2+lk(0y^l)2=(1y^k)2+lky^l2(2)=12y^k+y^l2

Given (2), it's straightforward to see that (1) indeed holds because only 2y^k depends on k.

Remark

The assumption k=1Ky^k=1 is actually not required.