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Let $x_1,\ldots,x_N$ be $N$ samples drawn independently from the normal distribution $\mathcal{N}(0,1)$. Let $\Delta_{ij} = | x_i - x_j |$ be the distance between the $i$-th and the $j$-th sample. Let $M_N = \min_{ij} \Delta_{ij}$ be the minimal distance (nearest neighbor distance) between all pairs of points.

Is the distribution of $M_N$ easy to compute?

I am particularly interested in making a statement of the form "$P(M_N \ge \mu) \ge 1 - \epsilon$", computing the limit $\mu$ for a given $\epsilon$.

The bound does not need to be tight, but it should not be catastrophically off. I am considering the non asymptotic case, $N$ will be between 1 and 10000, $\epsilon$ will be small (of the order of $10^{-6}$ to $10^{-12}$) -- which seems to hinder Monte Carlo computations.

(I have seen plenty of posts on MathOverflow and the Mathematics Stack Exchange about samples from the uniform distribution -- but nothing about normal variables.)

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This answer gives a bound of the requested form, but I have no idea whether it is "catastrophically off" :)

  • $\min_{ij} \Delta_{ij} < \mu \iff \text{some } \Delta_{ij} < \mu$

  • $P(\min_{ij} \Delta_{ij} < \mu) = P(\text{some } \Delta_{ij} < \mu) = P(\bigcup_{ij} (\Delta_{ij} < \mu)) \le \sum_{ij} P(\Delta_{ij} < \mu)$

  • $P(\min_{ij} \Delta_{ij} \ge \mu) = 1 - P(\min_{ij} \Delta_{ij} < \mu) > 1 - \sum_{ij} P(\Delta_{ij} < \mu) = 1 - \epsilon$

So we identify $\epsilon = \sum_{ij} P(\Delta_{ij} < \mu)$. But all the $x_i$ are i.i.d. $N(0,1)$, so any $(x_i - x_j) \sim N(0,\sqrt{2})$.

  • $\epsilon = \sum_{ij} P(\Delta_{ij} < \mu) = {N \choose 2} P(|N(0,\sqrt{2})| < \mu) = {N \choose 2} \text{ erf}({\mu \over 2}),$ where $\text{erf}$ is the error function

  • $\mu = 2 \text{ erf}^{-1} (\epsilon / {N \choose 2})$

For your typical values of $\epsilon < 10^{-6}$, especially if you then divide $\epsilon / {N \choose 2}$ for some large $N$, the argument to the error function is going to be tiny. In this case I think a reasonable approximation might be:

  • $x \approx 0 \implies \text{erf}(x) = {2 \over \sqrt{\pi}} \int^x_0 e^{-t^2} dt \approx {2 \over \sqrt{\pi}} \int^x_0 e^0 dt = {2 \over \sqrt{\pi}} x$

  • So $\mu \approx 2 {\sqrt{\pi} \over 2} \epsilon / {N \choose 2} = \color{red}{\sqrt{\pi} \epsilon / {N \choose 2}} $

Reminder: I have no idea how accurate / catastropically off this is. Hope this helps!

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