From Average Embeddings To Nearest Neighbor Search

TL;DR

This paper demonstrates how average embeddings can be used to develop efficient approximate nearest neighbor search algorithms, extending classic embedding methods with a data-dependent hashing approach.

Contribution

It introduces a novel approach leveraging average embeddings to improve approximate nearest neighbor search, strengthening traditional bi-Lipschitz embedding techniques.

Findings

Embedding metric spaces into  on average enables efficient approximate nearest neighbor search.

Existence of efficient average embeddings implies a polynomial approximation algorithm.

The approach enhances data-dependent hashing methods for similarity search.

Abstract

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.