From Average Embeddings To Nearest Neighbor Search
Andoni Alexandr, Cheikhi David. Arxiv 2021
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 \(ℓ₂\) 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.
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