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Efficient Set Similarity Estimation Using Min-wise Hashing Techniques

This supplemental material outlines min-wise hashing methods for estimating set similarity using the Jaccard measure. It presents the signature idea, which involves hashing sets’ columns to create compact signatures that approximate the similarity between sets. The document discusses key observations, such as the distribution of different row types in the matrix and the surprising properties of min-hashing. It also covers various implementation strategies, including random permutations and one-pass algorithms for efficient computation of signature matrices, enabling scalable similarity measurements between sets.

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Efficient Set Similarity Estimation Using Min-wise Hashing Techniques

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  1. CS276AText Information Retrieval, Mining, and Exploitation Supplemental Min-wise Hashing Slides [Brod97,Brod98] (Adapted from Rajeev Motwani’s CS361A slides)

  2. Set Similarity • Set Similarity (Jaccard measure) • View sets as columns of a matrix; one row for each element in the universe. aij = 1 indicates presence of item i in set j • Example C1C2 0 1 1 0 1 1 simJ(C1,C2) = 2/5 = 0.4 0 0 1 1 0 1

  3. Identifying Similar Sets? • Signature Idea • Hash columns Ci to signature sig(Ci) • simJ(Ci,Cj) approximated by simH(sig(Ci),sig(Cj)) • Naïve Approach • Sample P rows uniformly at random • Define sig(Ci) as P bits of Ci in sample • Problem • sparsity  could easily miss interesting part of columns • sample would get only 0’s in columns

  4. Key Observation • For columns Ci, Cj, four types of rows Ci Cj A 1 1 B 1 0 C 0 1 D 0 0 • Overload notation: A = # of rows of type A • Claim

  5. Min Hashing • Randomly permute rows • Hash h(Ci) = index of first row with 1 in column Ci • Suprising Property • Why? • Both are A/(A+B+C) • Look down columns Ci, Cj until first non-Type-D row • h(Ci) = h(Cj)  type A row

  6. Min-Hash Signatures • Pick – P random row permutations • MinHash Signature sig(C) = list of P indexes of first rows with 1 in column C • Similarity of signatures • Let simH(sig(Ci),sig(Cj)) = fraction of permutations where MinHash values agree • Observe E[simH(sig(Ci),sig(Cj))] = simJ(Ci,Cj)

  7. Example Signatures S1 S2 S3 Perm 1 = (12345) 1 2 1 Perm 2 = (54321) 4 5 4 Perm 3 = (34512) 3 5 4 C1 C2 C3 R1 1 0 1 R2 0 1 1 R3 1 0 0 R4 1 0 1 R5 0 1 0 Similarities 1-2 1-3 2-3 Col-Col 0.00 0.50 0.25 Sig-Sig 0.00 0.67 0.00 Exercise: What similarities do we get if we also picked a min-hash for Perm 4=(21453) ?

  8. Implementation Trick • Permuting rows even once is prohibitive • Row Hashing • Pick P hash functions hk: {1,…,n}{1,…,O(n)} • Ordering under hk gives random row permutation • One-pass Implementation • For each Ci and hk, keep “slot” for min-hash value • Initialize all slot(Ci,hk) to infinity • Scan rows in arbitrary order looking for 1’s • Suppose row Rj has 1 in column Ci • For each hk, • if hk(j) < slot(Ci,hk), then slot(Ci,hk)  hk(j)

  9. Example C1 C2 R1 1 0 R2 0 1 R3 1 1 R4 1 0 R5 0 1 C1 slotsC2 slots h(1) = 1 1 - g(1) = 3 3 - h(2) = 2 1 2 g(2) = 0 3 0 h(3) = 3 1 2 g(3) = 2 2 0 h(4) = 4 1 2 g(4) = 4 2 0 h(x) = x mod 5 g(x) = 2x+1 mod 5 h(5) = 0 1 0 g(5) = 1 2 0

  10. Comparing Signatures • Signature Matrix S • Rows = Hash Functions • Columns = Columns • Entries = Signatures • Compute – Pair-wise similarity of signature columns

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