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# In Defense of Minhash over Simhash

*Proceedings of the Seventeenth International Conference on Artificial Intelligence and Statistics*, PMLR 33:886-894, 2014.

#### Abstract

MinHash and SimHash are the two widely adopted Locality Sensitive Hashing (LSH) algorithms for large-scale data processing applications. Deciding which LSH to use for a particular problem at hand is an important question, which has no clear answer in the existing literature. In this study, we provide a theoretical answer (validated by experiments) that MinHash virtually always outperforms SimHash when the data are binary, as common in practice such as search. The collision probability of MinHash is a function of \em resemblance similarity (\mathcalR), while the collision probability of SimHash is a function of \em cosine similarity (\mathcalS). To provide a common basis for comparison, we evaluate retrieval results in terms of \mathcalS for both MinHash and SimHash. This evaluation is valid as we can prove that MinHash is a valid LSH with respect to \mathcalS, by using a general inequality \mathcalS^2≤\mathcalR≤\frac\mathcalS2-\mathcalS. Our \textbfworst case analysis can show that MinHash significantly outperforms SimHash in \textbfhigh similarity region. Interestingly, our intensive experiments reveal that MinHash is also substantially better than SimHash even in datasets where most of the data points are not too similar to each other. This is partly because, in practical data, often \mathcalR≥\frac\mathcalSz-\mathcalS holds where z is only slightly larger than 2 (e.g., z≤2.1). Our \textbfrestricted worst case analysis by assuming \frac\mathcalSz-\mathcalS≤\mathcalR≤\frac\mathcalS2-\mathcalS shows that MinHash indeed significantly outperforms SimHash even in \textbflow similarity region. We believe the results in this paper will provide valuable guidelines for search in practice, especially when the data are sparse.