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Latent Semantic Indexing (mapping onto a smaller space of latent concepts) Paolo Ferragina Dipartimento di Informatica Università di Pisa Reading 18

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Speeding up cosine computation What if we could take our vectors and “pack” them into fewer dimensions (say 50,000 100) while preserving distances? Now, O(nm) Then, O(km+kn) where k << n,m Two methods: “Latent semantic indexing” Random projection

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A sketch LSI is data-dependent Create a k-dim subspace by eliminating redundant axes Pull together “related” axes – hopefully car and automobile Random projection is data-independent Choose a k-dim subspace that guarantees good stretching properties with high probability between pair of points. What about polysemy ?

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Notions from linear algebra Matrix A, vector v Matrix transpose (A t ) Matrix product Rank Eigenvalues and eigenvector v: Av = v

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Overview of LSI Pre-process docs using a technique from linear algebra called Singular Value Decomposition Create a new (smaller) vector space Queries handled (faster) in this new space

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Singular-Value Decomposition Recall m n matrix of terms docs, A. A has rank r m,n Define term-term correlation matrix T=AA t T is a square, symmetric m m matrix Let P be m r matrix of eigenvectors of T Define doc-doc correlation matrix D=A t A D is a square, symmetric n n matrix Let R be n r matrix of eigenvectors of D

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A’s decomposition Do exist matrices P (for T, m r) and R (for D, n r) formed by orthonormal columns (unit dot-product) It turns out that A = P R t Where is a diagonal matrix with the eigenvalues of T=AA t in decreasing order. = A P RtRt mnmnmrmr rrrr rnrn

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For some k << r, zero out all but the k biggest eigenvalues in [choice of k is crucial] Denote by k this new version of , having rank k Typically k is about 100, while r ( A’s rank ) is > 10,000 = P kk RtRt Dimensionality reduction AkAk document useless due to 0-col/0-row of k m x r r x n r k k k 00 0 A m x k k x n

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Guarantee A k is a pretty good approximation to A: Relative distances are (approximately) preserved Of all m n matrices of rank k, A k is the best approximation to A wrt the following measures: min B, rank(B)=k ||A-B|| 2 = ||A-A k || 2 = k min B, rank(B)=k ||A-B|| F 2 = ||A-A k || F 2 = k 2 k+2 2 r 2 Frobenius norm ||A|| F 2 = 2 2 r 2

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Reduction X k = k R t is the doc-matrix k x n, hence reduced to k dim Take the doc-correlation matrix: It is D=A t A =(P R t ) t (P R t ) = ( R t ) t ( R t ) Approx with k, thus get A t A X k t X k (both are n x n matr.) We use X k to define how to project A and Q: X k = k R t, substitute R t = P t A, so get P k t A. In fact, k P t = P k t which is a k x m matrix This means that to reduce a doc/query vector is enough to multiply it by P k t Cost of sim(q,d), for all d, is O(kn+km) instead of O(mn) R,P are formed by orthonormal eigenvectors of the matrices D,T

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Which are the concepts ? c-th concept = c-th row of P k t (which is k x m) Denote it by P k t [c], whose size is m = #terms P k t [c][i] = strength of association between c-th concept and i-th term Projected document: d’ j = P k t d j d’ j [c] = strenght of concept c in d j Projected query: q’ = P k t q q’ [c] = strenght of concept c in q

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Random Projections Paolo Ferragina Dipartimento di Informatica Università di Pisa Slides only !

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An interesting math result f() is called JL-embedding Setting v=0 we also get a bound on f(u)’s stretching!!! Lemma (Johnson-Linderstrauss, ‘82) Let P be a set of n distinct points in m-dimensions. Given > 0, there exists a function f : P IR k such that for every pair of points u,v in P it holds: (1 - ) ||u - v|| 2 ≤ ||f(u) – f(v)|| 2 ≤ (1 + ) ||u-v|| 2 Where k = O( -2 log m)

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What about the cosine-distance ? f(u)’s, f(v)’s stretching substituting formula above

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How to compute a JL-embedding? E[r i,j ] = 0 Var[r i,j ] = 1 If we set R = r i,j to be a random mx k matrix, where the components are independent random variables with one of the following distributions

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Finally... Random projections hide large constants k (1/ ) 2 * log m, so it may be large… it is simple and fast to compute LSI is intuitive and may scale to any k optimal under various metrics but costly to compute

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Document duplication (exact or approximate) Paolo Ferragina Dipartimento di Informatica Università di Pisa Slides only!

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Duplicate documents The web is full of duplicated content Few exact duplicate detection Many cases of near duplicates E.g., Last modified date the only difference between two copies of a page Sec. 19.6

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Natural Approaches Fingerprinting: only works for exact matches, slow Checksum – no worst-case collision probability guarantees MD5 – cryptographically-secure string hashes Edit-distance metric for approximate string-matching expensive – even for one pair of strings impossible – for 10 32 web documents Random Sampling sample substrings (phrases, sentences, etc) hope: similar documents similar samples But – even samples of same document will differ

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Obvious techniques Checksum – no worst-case collision probability guarantees MD5 – cryptographically-secure string hashes relatively slow Karp-Rabin’s Scheme Rolling hash: split doc in many pieces Algebraic technique – arithmetic on primes Efficient and other nice properties… Exact-Duplicate Detection

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Karp-Rabin Fingerprints Consider – m-bit string A = 1 a 1 a 2 … a m Basic values: Choose a prime p in the universe U, such that 2p uses few memory-words (hence U ≈ 2 64 ) Fingerprints: f(A) = A mod p Nice property is that if B = a 2 … a m a m+1 f(B) = [2 m-1 (A – 2 m - a 1 2 m-1 ) + 2 m + a m+1 ] mod p Prob[false hit] = Prob p divides (A-B) = #div(A-B)/ #prime(U) ≈ (log (A+B)) / #prime(U) = m log U/U

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Near-Duplicate Detection Problem Given a large collection of documents Identify the near-duplicate documents Web search engines Proliferation of near-duplicate documents Legitimate – mirrors, local copies, updates, … Malicious – spam, spider-traps, dynamic URLs, … Mistaken – spider errors 30% of web-pages are near-duplicates [1997]

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Desiderata Storage: only small sketches of each document. Computation: the fastest possible Stream Processing : once sketch computed, source is unavailable Error Guarantees problem scale small biases have large impact need formal guarantees – heuristics will not do

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Basic Idea [Broder 1997] Shingling dissect document into q-grams (shingles) represent documents by shingle-sets reduce problem to set intersection [ Jaccard ] They are near-duplicates if large shingle-sets intersect enough

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Similarity of Documents Doc B SBSB SASA Doc A Jaccard measure – similarity of S A, S B Claim: A & B are near-duplicates if sim(S A,S B ) is high

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Basic Idea [Broder 1997] Shingling dissect document into q-grams (shingles) represent documents by shingle-sets reduce problem to set intersection [ Jaccard ] They are near-duplicates if large shingle-sets intersect enough We need to cope with “Set Intersection” fingerprints of shingles (for space/time efficiency) min-hash to estimate intersections sizes (for further time and space efficiency)

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Multiset of Fingerprints Doc shingling Multiset of Shingles fingerprint Documents Sets of 64-bit fingerprints Fingerprints: Use Karp-Rabin fingerprints over q-gram shingles (of 8q bits) Fingerprint space [0, …, U-1] In practice, use 64-bit fingerprints, i.e., U=2 64 Prob[collision] ≈ (8q)/2 64 << 1 This reduces space for storing the multi-sets And the time to intersect them, but...

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Speeding-up: Sketch of a document Intersecting shingle-sets is too costly Create a “sketch vector” (of size ~200) for each document, for its shingle-set Documents that share ≥ t (say 80%) corresponding vector elements are near duplicates Sec. 19.6

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Sketching by Min-Hashing Consider S A, S B P Pick a random permutation π of P (such as ax+b mod |P|) Define = π -1 ( min{π(S A )} ), = π -1 ( min{π(S B )} ) minimal element under permutation π Lemma:

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Strengthening it… Similarity sketch sk(A) = k minimal elements under π(S A ) K is fixed or is a fixed ratio of S A,S B ? We might also take K permutations and the min of each Similarity Sketches sk(A): Succinct representation of fingerprint sets S A Allows efficient estimation of sim(S A,S B ) Basic idea is to use min-hash of fingerprints Note : we can reduce the variance by using a larger k

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Computing Sketch[i] for Doc1 Document 1 2 64 Start with 64-bit f(shingles) Permute on the number line with i Pick the min value Sec. 19.6

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Test if Doc1.Sketch[i] = Doc2.Sketch[i] Document 1 Document 2 2 64 Are these equal? Test for 200 random permutations: , ,… 200 AB Sec. 19.6

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However… Document 1 Document 2 2 64 A = B iff the shingle with the MIN value in the union of Doc1 and Doc2 is common to both (i.e., lies in the intersection) Claim: This happens with probability Size_of_intersection / Size_of_union B A Sec. 19.6

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Sum up… Brute-force: compare sk(A) vs. sk(B) for all the pairs of documents A and B. Locality sensitive hashing (LSH) Compute sk(A) for each document A Use LSH of all sketches, briefly: Take h elements of sk(A) as ID (may induce false positives) Create t IDs (to reduce the false negatives) If one ID matches with another one (wrt same h-selection), then the corresponding docs are probably near-duplicates; hence compare.

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Search Engines “Semantic” searches ?

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“ Diego Maradona won against Mexico ” Dictionary of terms against Diego Maradona Mexico won 2.2 5.1 9.1 1.0 0.1 Term Vector Similarity(v,w) ≈ cos( ) t1t1 v w t3t3 t2t2 Vector Space model Classical approach Mainly term-based : polysemy and synonymy issues

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38 A new approach: Massive graphs of entities and relations May 2012

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the paparazzi photographed the star the astronomer photographed the star A typical issue: polysemy

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He is using Microsoft’s browser He is a fan of Internet Explorer Another issue: synonymy

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http://tagme.di.unipi.it

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Korean won Win-loss record Only won... “ Diego Maradona won against Mexico ” Diego A. Maradona Diego Maradona jr. Maradona Stadium Maradona Film … Mexico nation Mexico state Mexico football team Mexico baseball team … No Annotation PARSING PRUNING 2 simple features DISAMBIGUATION by a voting scheme TAGME score

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obama asks iran for RQ-170 sentinel drone back us president issues Ahmadinejad ultimatum Barack Obama Iran Lockheed Martin RQ-170 Sentinel President of the United States Mahmoud Ahmadinejad Ultimatum Why is it more powerful ?

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44 Text as a sub-graph of topics Mahmoud Ahmadinejad Ultimatum RQ-170 drone President of the United States Barack Obama Iran

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Text as a sub-graph of topics Mahmoud Ahmadinejad Ultimatum RQ-170 drone Any relatedness measure over a graph, e.g. [Milne & Witten, 2008] President of the United States Barack Obama Iran Graph analysis allows to find similarities between texts and entities even if they do not match syntactically (so at concept level)

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Search Results Clustering Jaguar Cars Panthera Onca Mac OS X Atari Jaguar Jacksonville Jags Fender Jaguar … TOPICS

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Paper at ACM WSDM 2012 Paper at ECIR 2012 Paper at IEEE Software 2012 Pls design your killer app... http://acube.di.unipi.it/tagme Releasing open-source…

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