This is an informal discussion about the Kn-Nearest-Neighbor (KNN) and Nearest Neighbor (NN) estimations in k = 1 case. My conclusion is that KNN and NN are essentially the same when k = 1.
In any testing point, let us assume that this testing point is category-k dominant using NN (trainning point from category k will be firstly captured). Then, if we switch to use KNN instead, when enlarge the volume surrounding this testing point, the volume in category k should be the smallest (with largest discriminant value), so this point is still considered as category-k dominant using KNN.
Based on the above discussion, I conclude that the classification using KNN and NN when k = 1 should be the same.
zge
