Theory of Locality Sensitive Hashing - SNAP - Stanford University

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Prob. h π(C 1) = h π(C 2) is the same as Sim(D 1, D 2): Pr[h π(C 1) = h π(C 2)] = Sim(D 1, D 2) Permutation π 1 3 7 6 2 5 4 4 2 1 3 6 7 5 3 4 7 6 1 2 5 1 1 0 0 0 1 1 Input matrix 0 0 1 1 1 0 0 1 0 0 0 0 1 1 0 1 1 1 1 0 0 Signature matrix M 1/20/2012 Jure Leskovec, Stanford C246: Mining Massive Datasets 6 2 1 2 1 2 1 4 1 2 1 2 1
ands Hash cols of signature matrix M: Similar columns likely hash to same bucket r rows Divide matrix M into b bands of r rows Candidate column pairs are those that hash to the same bucket for ≥ 1 band Buckets Matrix M Prob. of sharing a bucket Sim. threshold s 1/20/2012 Jure Leskovec, Stanford C246: Mining Massive Datasets 7 2 1 2 1 2 1 4 1 2 1 2 1
Page 1 and 2: CS246: Mining Massive Datasets Jure
Page 3 and 4: 1. Shingling: Convert documents to
Page 5: A k-shingle (or k-gram) is a sequen
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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