Fast Similarity Sketching

We consider the $\textit{Similarity Sketching}$ problem: Given a universe $[u] = \{0,\ldots, u-1\}$ we want a random function $S$ mapping subsets $A\subseteq [u]$ into vectors $S(A)$ of size $t$, such that the Jaccard similarity $J(A,B) = |A\cap B|/|A\cup B|$ between sets $A$ and $B$ is preserved. More precisely, define $X_i = [S(A)[i] = S(B)[i]]$ and $X = \sum_{i\in [t]} X_i$. We want $E[X_i]=J(A,B)$, and we want $X$ to be strongly concentrated around $E[X] = t \cdot J(A,B)$ (i.e. Chernoff-style bounds). This is a fundamental problem which has found numerous applications in data mining, large-scale classification, computer vision, similarity search, etc. via the classic MinHash algorithm. The vectors $S(A)$ are also called $\textit{sketches}$. Strong concentration is critical, for often we want to sketch many sets $B_1,\ldots,B_n$ so that we later, for a query set $A$, can find (one of) the most similar $B_i$. It is then critical that no $B_i$ looks much more similar to $A$ due to errors in the sketch. The seminal $t\times\textit{MinHash}$ algorithm uses $t$ random hash functions $h_1,\ldots, h_t$, and stores $\left ( \min_{a\in A} h_1(A),\ldots, \min_{a\in A} h_t(A) \right )$ as the sketch of $A$. The main drawback of MinHash is, however, its $O(t\cdot |A|)$ running time, and finding a sketch with similar properties and faster running time has been the subject of several papers. (continued...)