New Instability Results for High Dimensional Nearest Neighbor Search

Consider a dataset of n(d) points generated independently from R according to a common p.d.f. fd with support(fd) = [0, 1] d and sup{fd([0, 1] )} growing sub-exponentially in d. We prove that: (i) if n(d) grows sub-exponentially in d, then, for any query point ~q ∈ [0, 1] and any ǫ > 0, the ratio of the distance between any two dataset points and ~q is less that 1 + ǫ with probability → 1 as d → ∞; (ii) if n(d) > [4(1 + ǫ)] for large d, then for all ~q ∈ [0, 1] (except a small subset) and any ǫ > 0, the distance ratio is less than 1 + ǫ with limiting probability strictly bounded away from one. Moreover, we provide preliminary results along the lines of (i) when fd = N(~μd,Σd).