It is well known that the $p$-stable distribution can be used to generate locality sensitive hash code for $\ell_p$-norm distance measure where $p \le 2$. However, it seems that the situation for $p>2$ is not clear. Does there exist a locality sensitive hashing for $\ell_p$-norm distance where $p>2$? If not, what does it happen at the specific value $p=2$ that forms a "phase transition"?


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