Theory of Locality Sensitive Hashing - SNAP - Stanford University

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Docu- ment The set of strings of length k that appear in the document Signatures : short integer vectors that represent the sets, and reflect their similarity 1/20/2012 Jure Leskovec, Stanford C246: Mining Massive Datasets Locality- sensitive Hashing Candidate pairs : those pairs of signatures that we need to test for similarity. 4
A k-shingle (or k-gram) is a sequence of k tokens that appears in the document Example: k=2; D 1= abcab Set of 2-shingles: C 1 = S(D 1) = {ab, bc, ca} Represent a doc by the set of hash values of its k-shingles A natural document similarity measure is then the Jaccard similarity: Sim(D 1, D 2) = |C 1∩C 2|/|C 1∪C 2| 1/20/2012 Jure Leskovec, Stanford C246: Mining Massive Datasets 5
Page 1 and 2: CS246: Mining Massive Datasets Jure
Page 3: 1. Shingling: Convert documents to
Page 7 and 8: ands Hash cols of signature matrix
Page 9 and 10: Remember: b bands, r rows/band Col
Page 11 and 12: Docu- ment The set of strings of le
Page 13 and 14: For min-hash signatures, we got a m
Page 15 and 16: High probability; at least p 1 Pr[h
Page 17 and 18: Claim: H is a (1/3, 2/3, 2/3, 1/3)-
Page 19 and 20: Given family H, construct family H
Page 21 and 22: AND makes all probs. shrink, but by
Page 23 and 24: Picking r and b to get the best 50
Page 25 and 26: p 1-(1-p 4 ) 4 .2 .0064 .3 .0320 .4
Page 27 and 28: p (1-(1-p) 4 ) 4 .1 .0140 .2 .1215
Page 29 and 30: Pick any two distances x < y Start
Page 31 and 32: For cosine distance, d(A, B) = θ =
Page 33 and 34: Look in the plane of x and y. Hyper
Page 35 and 36: Expensive to pick a random vector i
Page 37 and 38: Bucket width a Points at distance d
Page 39 and 40: If points are distance d < a/2, pro
Page 41: But as long as x < y, the probabili

Theory of Locality Sensitive Hashing - SNAP - Stanford University

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