In the paper "When Is 'Nearest Neighbor' Meaningful?" we read that,
We show that under certain broad conditions (in terms of data and query distributions, or workload), as dimensionality increases, the distance to the nearest neighbor approaches the distance to the farthest neighbor. In other words, the contrast in distances to different data points becomes nonexistent. The conditions we have identied in which this happens are much broader than the independent and identically distributed (IID) dimensions assumption that other work assumes.
My question is, how I should generate a dataset that produces this effect?
I have created three points each with 1000 dimensions with random numbers ranging from 0-255 for each dimension but points create different distances and do not reproduce what is mentioned above. It seems changing dimensions (e.g. 10 or 100 or 1000 dimensions) and ranges (e.g. [0,1]) do not change anything. I still get different distances which should not be any problem for e.g. clustering algorithms!
Edit: I have tried more samples, based on my experiments distances between points do not converge to any number, on the contrary the max and min distances between points get more apparent. This is also in contrary to what is written in the first post of Need more intuition for the curse of dimensionality and also many other places which claim the same thing like https://en.wikipedia.org/wiki/Clustering_high-dimensional_data#Problems. I would still appreciate it if someone can show me with a piece of code or real dataset that such effect exist in practical scenarios.