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In this work we study the validity of the so-called curse of dimensionalityfor indexing of databases for similarity search. We perform an asymptoticanalysis, with a test model based on a sequence of metric spaces $(\Omega_d)$from which we pick datasets $X_d$ in an i.i.d. fashion. We call the subscript$d$ the dimension of the space $\Omega_d$ (e.g. for $\mathbb{R}^d$ thedimension is just the usual one) and we allow the size of the dataset $n=n_d$to be such that $d$ is superlogarithmic but subpolynomial in $n$.  We study the asymptotic performance of pivot-based indexing schemes where thenumber of pivots is $o(n/d)$. We pick the relatively simple cost model ofsimilarity search where we count each distance calculation as a singlecomputation and disregard the rest.  We demonstrate that if the spaces $\Omega_d$ exhibit the (fairly common)concentration of measure phenomenon the performance of similarity search usingsuch indexes is asymptotically linear in $n$. That is for large enough $d$ thedifference between using such an index and performing a search without an indexat all is negligeable. Thus we confirm the curse of dimensionality in thissetting.

Curse of Dimensionality in the Application of Pivot-based Indexes to the Similarity Search Problem