1-NN learning on Hanan sets

The output of the nearest neighbor (1-NN) classification rule, g_S,q(x), depends on a given learning set S_N and on a distance function ρ_q(x,X).

We show that transforming S_{N} into a set A_{N} whose patterns have a Hanan grid-like structure, results in the equivalence g_A,q(x) = g_A,p(x) that holds for any NN classifier with distance functions ‖x-X‖_q and with any q ∈ (0,∞). Thanks to the equivalence, A_N can be used to learn g_A,q(x) to mimic a behavior of the classifier g_S,p(x) based on the original set S_N even when q is unknown (and varying).

Possible application of the proposed framework (inspired also by a time-varying stimuli perception phenomenon) in autism spectrum disorder (ASD) therapeutic tools design is discussed.