Distance contrast and the failure of nearest neighbours

ProofIntermediate⏱ 30 minClassical MLknncontrastconcentrationcurse

Theory: Nearest Neighbours and the Curse of Dimensionality →

Problem

Using norm concentration, show that for $N$ i.i.d. isotropic points in $\R^d$ the ratio of the farthest to the nearest distance from a query tends to $1$ as $d\to\infty$ with $N$ fixed. Then explain carefully why nearest-neighbour search nevertheless works on real high-dimensional data.

Write a complete argument. Every inequality you use should be one you could state precisely.

Solution

Opening this before you have a real attempt on paper is the fastest way to feel like you understand something you cannot reproduce.

Related problems

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Simulate distance contrast run	Classical ML	Coding	Beginner	30 min
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Simulate the measure of a spherical cap run	High-Dim Probability	Coding	Intermediate	30 min
Distance contrast in high dimension	High-Dim Probability	Proof	Advanced	1 hour