  In this note, we show that one can use average embeddings, introduced
recently in [Naor'20, arXiv:1905.01280], to obtain efficient algorithms for
approximate nearest neighbor search. In particular, a metric $X$ embeds into
$\ell_2$ on average, with distortion $D$, if, for any distribution $\mu$ on
$X$, the embedding is $D$ Lipschitz and the (square of) distance does not
decrease on average (wrt $\mu$). In particular existence of such an embedding
(assuming it is efficient) implies a $O(D^3)$ approximate nearest neighbor
search under $X$. This can be seen as a strengthening of the classic
(bi-Lipschitz) embedding approach to nearest neighbor search, and is another
application of data-dependent hashing paradigm.
