Theory of Locality Sensitive Hashing - SNAP - Stanford University

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Pick a random vector v, which determines a hash function h v with two buckets h v(x) = +1 if v⋅x > 0; = -1 if v⋅x < 0 LS-family H = set of all functions derived from any vector Claim: For points x and y, Pr[h(x) = h(y)] = 1 – d(x,y)/180 1/19/2011 Jure Leskovec, Stanford C246: Mining Massive Datasets 32
Look in the plane of x and y. Hyperplane normal to v h(x) = h(y) v θ 1/19/2011 Jure Leskovec, Stanford C246: Mining Massive Datasets 33 x y Prob[Red case] = θ/180 v Hyperplane normal to v h(x) ≠ h(y)
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 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: For cosine distance, d(A, B) = θ =
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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