On the geometry of similarity search: dimensionality curse and concentration of measure

We suggest that the curse of dimensionality affecting the similarity-based search in large datasets is a manifestation of the phenomenon of concentration of measure on high-dimensional structures. We prove that, under certain geometric assumptions on the query domain $\mathrm{\Omega }$ and the dataset $X$, if $\mathrm{\Omega }$ satisfies the so-called concentration property, then for most query points $x^{\ast }$ the ball of radius $(1+\text{e})d_X(x^{\ast })$ centred at $x^{\ast }$ contains either all points of $X$ or else at least $C_1\exp (-C_2{\text{e}}^{2}n)$ of them. Here $d_X(x^{\ast })$ is the distance from $x^{\ast }$ to the nearest neighbour in $X$ and $n$ is the dimension of $\mathrm{\Omega }$.