[ohxh]

Johnson-lindenstrauss lemma [1] for anyone curious. But you can only map to k>8(\ln N)/\varepsilon ^{2}} if you want to preserve distances within a factor of \varepsilon with a JL-transform. This is tight up to a constant factor too.

I always wondered: if we want to preserve distances between a billion points within 10%, that would mean we need ~18k dimensions. 1% would be 1.8m. Is there a stronger version of the lemma for points that are well spread out? Or are embeddings really just fine with low precision for the distance?

[1] https://en.wikipedia.org/wiki/Johnson%E2%80%93Lindenstrauss_...