Theory of Locality Sensitive Hashing - SNAP - Stanford University

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Can we reproduce the “S-curve” effect we saw before for any LS family? Prob. of sharing a bucket The “bands” technique we learned for signature matrices carries over to this more general setting Two constructions: AND construction like “rows in a band” OR construction like “many bands” Similarity s 1/19/2011 Jure Leskovec, Stanford C246: Mining Massive Datasets 18
Given family H, construct family H’ consisting of r functions from H For h = [h 1,…,h r] in H’, h(x)=h(y) if and only if h i(x)=h i(y) for all i Theorem: If H is (d 1,d 2,p 1,p 2)-sensitive, then H’ is (d 1,d 2,(p 1) r ,(p 2) r )-sensitive Proof: Use the fact that h i ’s are independent 1/19/2011 Jure Leskovec, Stanford C246: Mining Massive Datasets 19
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: Claim: H is a (1/3, 2/3, 2/3, 1/3)-
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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