Theory of Locality Sensitive Hashing - SNAP - Stanford University

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-way AND followed by b-way OR construction Exactly what we did with min-hashing If bands match in all r values hash to same bucket Cols that are hashed into at least 1 common bucket Candidate Take points x and y s.t. Pr[h(x) = h(y)] = p H will make (x,y) a candidate pair with prob. p Construction makes (x,y) a candidate pair with probability 1-(1-p r ) b The S-Curve! Example: Take H and construct H’ by the AND construction with r = 4. Then, from H’, construct H’’ by the OR construction with b = 4 1/19/2011 Jure Leskovec, <strong>Stanford</strong> C246: Mining Massive Datasets 22
Picking r and b to get the best 50 hash-functions (r=5, b=10) Prob. sharing a bucket 1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Similarity Blue area: False Negative rate Green area: False Positive rate 1/19/2011 Jure Leskovec, <strong>Stanford</strong> C246: Mining Massive Datasets 23
Page 1 and 2: CS246: Mining Massive Datasets Jure
Page 3 and 4: 1. Shingling: Convert documents to
Page 5 and 6: A k-shingle (or k-gram) is a sequen
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 makes all probs. shrink, but by
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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