On The Geometry Of Similarity Search Dimensionality Curse And Concentration Of Measure

Pestov Vladimir. Information Processing Letters 1999

[Paper]    
NEURIPS

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 $\Omega$ and the dataset $X$, if $\Omega$ 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 $\Omega$.

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