Piazza Recitation session : Review of linear algebra Location: - PowerPoint PPT Presentation

Piazza Recitation session : ¡ Review of linear algebra § Location: Thursday, April 11, from 3:30-5:20 pm in SIG 134 (here) Deadlines next Thu, 11:59 PM : ¡ HW0, HW1 How to find teammates for project? ¡ Piazza Team Search ¡ Make sure you have a good dataset accessible 4/10/19 Tim Althoff, UW CS547: Machine Learning for Big Data, http://www.cs.washington.edu/cse547 1

CS246: Mining Massive Datasets Jure Leskovec, Stanford University http://cs246.stanford.edu

¡ Task: Given a large number ( N in the millions or billions) of documents, find “near duplicates” ¡ Problem: § Too many documents to compare all pairs ¡ Solution: Hash documents so that similar documents hash into the same bucket § Documents in the same bucket are then candidate pairs whose similarity is then evaluated 4/10/19 Tim Althoff, UW CS547: Machine Learning for Big Data, http://www.cs.washington.edu/cse547 3

Candidate pairs: Locality- those pairs M i n - H a s h - Docu- S h i n g l i n g sensitive i n g of signatures ment Hashing that we need to test for similarity The set Signatures: of strings short integer of length k vectors that that appear represent the in the doc- sets, and ument reflect their similarity 4/10/19 Tim Althoff, UW CS547: Machine Learning for Big Data, http://www.cs.washington.edu/cse547 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 a set of hash values of its k -shingles ¡ A natural similarity measure is then the Jaccard similarity: sim (D 1 , D 2 ) = |C 1 Ç C 2 |/|C 1 È C 2 | § Similarity of two documents is the Jaccard similarity of their shingles 4/10/19 Tim Althoff, UW CS547: Machine Learning for Big Data, http://www.cs.washington.edu/cse547 6

¡ Min-Hashing : Convert large sets into short signatures, while preserving similarity: Pr[ h (C 1 ) = h (C 2 )] = sim (D 1 , D 2 ) Permutation p Input matrix (Shingles x Documents) Signature matrix M 2 4 3 1 0 1 0 2 1 2 1 1 0 0 1 3 2 4 2 1 4 1 0 1 0 1 7 1 7 1 2 1 2 0 1 0 1 6 3 2 Similarities of columns and 0 1 0 1 1 6 6 signatures (approx.) match! 1-3 2-4 1-2 3-4 5 7 1 1 0 1 0 Col/Col 0.75 0.75 0 0 Sig/Sig 0.67 1.00 0 0 4 5 5 1 0 1 0 4/10/19 Tim Althoff, UW CS547: Machine Learning for Big Data, http://www.cs.washington.edu/cse547 7

¡ Hash columns of the signature matrix M: Similar columns likely hash to same bucket § Divide matrix M into b bands of r rows (M=b·r) § Candidate column pairs are those that hash to the same bucket for ≥ 1 band Buckets Prob. of sharing Threshold s ≥ 1 bucket b bands r rows Similarity Matrix M 4/10/19 Tim Althoff, UW CS547: Machine Learning for Big Data, http://www.cs.washington.edu/cse547 8

Signatures: short Candidate pairs: integer signatures that those pairs of Locality- reflect point similarity H a s h signatures that sensitive Points f u n c . we need to test Hashing for similarity Design a locality sensitive Apply the hash function (for a given “Bands” technique distance metric) 4/10/19 Tim Althoff, UW CS547: Machine Learning for Big Data, http://www.cs.washington.edu/cse547 9

¡ The S-curve is where the “magic” happens Remember: Probability of sharing Threshold s Probability of Probability=1 equal hash-values ≥ 1 bucket = similarity if t>s No chance if t<s Similarity t of two sets Similarity t of two sets This is what 1 hash-code gives you This is what we want! Pr[ h p (C 1 ) = h p (C 2 )] = s im (D 1 , D 2 ) How to get a step-function? By choosing r and b ! 4/10/19 Tim Althoff, UW CS547: Machine Learning for Big Data, http://www.cs.washington.edu/cse547 10

¡ Remember: b bands, r rows/band ¡ Let sim( C 1 , C 2 ) = s What’s the prob. that at least 1 band is equal? ¡ Pick some band ( r rows) § Prob. that elements in a single row of columns C 1 and C 2 are equal = s § Prob. that all rows in a band are equal = s r § Prob. that some row in a band is not equal = 1 - s r ¡ Prob. that all bands are not equal = (1 - s r ) b ¡ Prob. that at least 1 band is equal = 1 - (1 - s r ) b P(C 1 , C 2 is a candidate pair) = 1 - (1 - s r ) b 4/10/19 Tim Althoff, UW CS547: Machine Learning for Big Data, http://www.cs.washington.edu/cse547 11

¡ Picking r and b to get the best S-curve § 50 hash-functions (r=5, b=10) 1 0.9 Prob. sharing a bucket 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, s 4/10/19 Tim Althoff, UW CS547: Machine Learning for Big Data, http://www.cs.washington.edu/cse547 12

1 1 r = 5, b = 1..50 Prob(Candidate pair) r = 1..10, b = 1 0.9 0.9 0.8 0.8 Given a fixed 0.7 0.7 0.6 0.6 threshold s . 0.5 0.5 0.4 0.4 0.3 0.3 We want choose 0.2 0.2 0.1 0.1 r and b such 0 0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 that the 1 1 r = 10, b = 1..50 Prob(Candidate pair) 0.9 0.9 P(Candidate 0.8 0.8 0.7 0.7 pair) has a 0.6 0.6 “step” right 0.5 0.5 0.4 0.4 around s . 0.3 0.3 0.2 0.2 r = 1, b = 1..10 0.1 0.1 0 0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Similarity Similarity prob = 1 - (1 - t r ) b 4/10/19 Tim Althoff, UW CS547: Machine Learning for Big Data, http://www.cs.washington.edu/cse547 13

Candidate pairs: Locality- those pairs M i n - H a s h - sensitive i n g of signatures Hashing that we need to test for similarity Signatures: short vectors that represent the sets, and reflect their similarity

¡ We have used LSH to find similar documents § More generally, we found similar columns in large sparse matrices with high Jaccard similarity ¡ Can we use LSH for other distance measures? § e.g., Euclidean distances, Cosine distance § Let’s generalize what we’ve learned! 4/10/19 Tim Althoff, UW CS547: Machine Learning for Big Data, http://www.cs.washington.edu/cse547 15

¡ d() is a distance measure if it is a function from pairs of points x,y to real numbers such that: § ! ", $ ≥ 0 § ! ", $ = 0 ()) " = $ § !(", $) = !($, ") § ! ", $ ≤ !(", -) + !(-, $) (triangle inequality) ¡ Jaccard distance for sets = 1 - Jaccard similarity ¡ Cosine distance for vectors = angle between the vectors ¡ Euclidean distances: § L 2 norm : d(x,y) = square root of the sum of the squares of the differences between x and y in each dimension § The most common notion of “distance” § L 1 norm : sum of absolute value of the differences in each dimension § Manhattan distance = distance if you travel along coordinates only 4/10/19 Tim Althoff, UW CS547: Machine Learning for Big Data, http://www.cs.washington.edu/cse547 16

¡ For Min-Hashing signatures, we got a Min-Hash function for each permutation of rows ¡ A “hash function” is any function that allows us to say whether two elements are “equal” § Shorthand: h(x) = h(y) means “ h says x and y are equal ” ¡ A family of hash functions is any set of hash functions from which we can pick one at random efficiently § Example: The set of Min-Hash functions generated from permutations of rows 4/10/19 Tim Althoff, UW CS547: Machine Learning for Big Data, http://www.cs.washington.edu/cse547 20

Suppose we have a space S of points with ¡ a distance measure d(x,y) Critical assumption A family H of hash functions is said to be ¡ ( d 1 , d 2 , p 1 , p 2 )- sensitive if for any x and y in S : 1. If d(x, y) < d 1 , then the probability over all h Î H , that h(x) = h(y) is at least p 1 2. If d(x, y) > d 2 , then the probability over all h Î H , that h(x) = h(y) is at most p 2 With a LS Family we can do LSH! 4/10/19 Tim Althoff, UW CS547: Machine Learning for Big Data, http://www.cs.washington.edu/cse547 21

Distance Small distance, Notice it’s a distance, not similarity, threshold t hence the S-curve is flipped! high probability p 1 Pr [ h (x) = h (y)] p 2 Large distance, low probability of hashing to the same value d 1 d 2 Distance d(x,y) 4/10/19 Tim Althoff, UW CS547: Machine Learning for Big Data, http://www.cs.washington.edu/cse547 22

¡ Let: § S = space of all sets, § d = Jaccard distance, § H is family of Min-Hash functions for all permutations of rows ¡ Then for any hash function h Î H : Pr[h(x) = h(y)] = 1 - d(x, y) § Simply restates theorem about Min-Hashing in terms of distances rather than similarities 4/10/19 Tim Althoff, UW CS547: Machine Learning for Big Data, http://www.cs.washington.edu/cse547 23

¡ Claim: Min-hash H is a (1/3, 2/3, 2/3, 1/3)- sensitive family for S and d . Then probability If distance < 1/3 that Min-Hash values (so similarity ≥ 2/3) agree is > 2/3 ¡ For Jaccard similarity, Min-Hashing gives a (d 1 ,d 2 ,(1-d 1 ),(1-d 2 ))- sensitive family for any d 1 <d 2 4/10/19 Tim Althoff, UW CS547: Machine Learning for Big Data, http://www.cs.washington.edu/cse547 24