Beyond Pairwise: Provably Fast Algorithms for Approximate $k$-Way Similarity Search

Part of Advances in Neural Information Processing Systems 26 (NIPS 2013)

Authors

Anshumali Shrivastava, Ping Li

Abstract

We go beyond the notion of pairwise similarity and look into search problems with $k$-way similarity functions. In this paper, we focus on problems related to \emph{3-way Jaccard} similarity: $\mathcal{R}^{3way}= \frac{|S_1 \cap S_2 \cap S_3|}{|S_1 \cup S_2 \cup S_3|}$, $S_1, S_2, S_3 \in \mathcal{C}$, where $\mathcal{C}$ is a size $n$ collection of sets (or binary vectors). We show that approximate $\mathcal{R}^{3way}$ similarity search problems admit fast algorithms with provable guarantees, analogous to the pairwise case. Our analysis and speedup guarantees naturally extend to $k$-way resemblance. In the process, we extend traditional framework of \emph{locality sensitive hashing (LSH)} to handle higher order similarities, which could be of independent theoretical interest. The applicability of $\mathcal{R}^{3way}$ search is shown on the Google sets" application. In addition, we demonstrate the advantage of $\mathcal{R}^{3way}$ resemblance over the pairwise case in improving retrieval quality."