New instability results for high-dimensional nearest neighbor search

Abstract

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