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

In this work we study the validity of the so-called curse of dimensionality for indexing of databases for similarity search. We perform an asymptotic analysis, with a test model based on a sequence of metric spaces (Ωd) from which we pick datasets Xd in an i.i.d. fashion. We call the subscript d the dimension of the space Ωd (e.g. for R d the dimension is just the usual one) and we allow the size of the dataset n = nd to be such that d is superlogarithmic but subpolynomial in n. We study the asymptotic performance of pivot-based indexing schemes where the number of pivots is o(n/d). We pick the relatively simple cost model of similarity search where we count each distance calculation as a single computation and disregard the rest. We demonstrate that if the spaces Ωd exhibit the (fairly common) concentration of measure phenomenon the performance of similarity search using such indexes is asymptotically linear in n. That is for large enough d the difference between using such an index and performing a search without an index at all is negligeable. Thus we confirm the curse of dimensionality in this setting.