Data-Dependent LSH for the Earth Mover's Distance

We give new data-dependent locality sensitive hashing schemes (LSH) for the Earth Mover's Distance ($\mathsf{EMD}$), and as a result, improve the best approximation for nearest neighbor search under $\mathsf{EMD}$ by a quadratic factor. Here, the metric $\mathsf{EMD}_s(\mathbb{R}^d,\ell_p)$ consists of sets of $s$ vectors in $\mathbb{R}^d$, and for any two sets $x,y$ of $s$ vectors the distance $\mathsf{EMD}(x,y)$ is the minimum cost of a perfect matching between $x,y$, where the cost of matching two vectors is their $\ell_p$ distance. Previously, Andoni, Indyk, and Krauthgamer gave a (data-independent) locality-sensitive hashing scheme for $\mathsf{EMD}_s(\mathbb{R}^d,\ell_p)$ when $p \in [1,2]$ with approximation $O(\log^2 s)$. By being data-dependent, we improve the approximation to $\tilde{O}(\log s)$. Our main technical contribution is to show that for any distribution $\mu$ supported on the metric $\mathsf{EMD}_s(\mathbb{R}^d, \ell_p)$, there exists a data-dependent LSH for dense regions of $\mu$ which achieves approximation $\tilde{O}(\log s)$, and that the data-independent LSH actually achieves a $\tilde{O}(\log s)$-approximation outside of those dense regions. Finally, we show how to "glue" together these two hashing schemes without any additional loss in the approximation. Beyond nearest neighbor search, our data-dependent LSH also gives optimal (distributional) sketches for the Earth Mover's Distance. By known sketching lower bounds, this implies that our LSH is optimal (up to $\mathrm{poly}(\log \log s)$ factors) among those that collide close points with constant probability.