We consider the problem of designing Locality Sensitive Filters (LSF)for set overlaps, also known as maximum inner product search on binary data. We give a simple data structure that generalizes and outperforms previous algorithms such as MinHash [J. Discrete Algorithms 1998], SimHash [STOC 2002], Spherical LSF [SODA  2017] and Chosen Path [STOC 2017] ; and we show matching lower bounds using hypercontractive inequalities for a wide range of parameters and space/time trade-offs. This answers the main open question in Christiani and Pagh STOC 2017%DIFDELCMD < ] %%% on unifying the landscape of Locality Sensitive (non-data-dependent) set similarity search in both time and space; or in the near-constant time regime, in space, by an arbitrarily large polynomial factor. Turning the geometric concept, based on Boolean supermajority functions, into a practical algorithm requires ideas from branching random walks on \mathbb Z^2, for which we give the first non-asymptotic near tight analysis. Our lower bounds follow from new hypercontractive arguments, which can be seen as characterizing the exact family of similarity search problems for which supermajorities are optimal. The optimality holds for among all hashing based data structures in the random setting, and by reductions, for 1 cell and 2 cell probe data structures. As a side effect, we obtain new hypercontractive bounds on the directed noise operator T^{p_1 \to p_2}_\rho} .
We formulate and optimally solve a new generalized Set Similarity Search problem, which assumes the size of the database and query sets are known in advance. By creating polylog copies of our data-structure, we optimally solve any symmetric Approximate Set Similarity Search problem, including approximate versions of Subset Search, Maximum Inner Product Search (MIPS), Jaccard Similarity Search and Partial Match. Our algorithm can be seen as a natural generalization of previous work on Set as well as Euclidean Similarity Search, but conceptually it differs by optimally exploiting the information present in the sets as well as their complements, and doing so asymmetrically between queries and stored sets. Doing so we improve upon the best previous work: MinHash [J. Discrete Algorithms 1998], SimHash [STOC 2002], Spherical LSF [SODA 2016, 2017] and Chosen Path [STOC 2017] %DIFDELCMD < ] %%% by as much as a factor n^{0.14 in both time and space; or in the near-constant time regime, in space, by an arbitrarily large polynomial factor. Turning the geometric concept, based on Boolean supermajority functions, into a practical algorithm requires ideas from branching random walks on \mathbb Z^2, for which we give the first non-asymptotic near tight analysis. Our lower bounds follow from new hypercontractive arguments, which can be seen as characterizing the exact family of similarity search problems for which supermajorities are optimal. The optimality holds for among all hashing based data structures in the random setting, and by reductions, for 1 cell and 2 cell probe data structures. As a side effect, we obtain new hypercontractive bounds on the directed noise operator T^{p_1 \to p_2}_\rho} .
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consider the problem of designing Locality Sensitive Filters (LSF)for set overlaps, also known as maximum inner product search on binary data. We give a simple data structure that generalizes and outperforms previous algorithms such as
formulate and optimally solve a new generalized Set Similarity Search problem, which assumes the size of the database and query sets are known in advance. By creating polylog copies of our data-structure, we optimally solve any symmetric Approximate Set Similarity Search problem, including approximate versions of Subset Search, Maximum Inner Product Search (MIPS), Jaccard Similarity Search and Partial Match. Our algorithm can be seen as a natural generalization of previous work on Set as well as Euclidean Similarity Search, but conceptually it differs by optimally exploiting the information present in the sets as well as their complements, and doing so asymmetrically between queries and stored sets. Doing so we improve upon the best previous work:
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; and we show matching lower bounds using hypercontractive inequalities for a wide range of parameters and space/time trade-offs. This answers the main open question in Christiani and Pagh
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STOC 2017
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on unifying the landscape of Locality Sensitive (non-data-dependent) set similarity search
by as much as a factor n^{0.14