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Text Databases
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Text Types
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Unstructured text
semi-structured text
structured text
Query: User wants to find documents related to a topic T
The search program tries to find the documents in the text
database that contain the string T
Two problems
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Synonymy: Given a word T, the word T does not occur anywhere
in a document D, even though D is in fact closely related to topic T
Polysemy: The same word may mean many different things in
different contexts
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We discuss,
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Measures of performance of a text retrieval system
Latent semantic indexing
Telescopic-Vector trees for document retrieval
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Precision and Recall
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Precision:
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How many of the returned documents are relevant?
(20+1)/(20+150+1)
Recall:
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How many of the relevant documents are returned?
(20+1)/(20+50+1)
50 20
150
Returned documents
Relevant documents
All documents
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Some Concepts
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Stop List
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Word Stems
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A set of words that do not “discriminate” between the documents
in a given archive
E.g.: Cornell SMART system has about 440 words on its stop list
Many words are small syntactic variants of each other
E.g., drug, drugged, drugs are similar in the sense that they share
a common “stem,” the word drug
Most document retrieval systems first eliminate words on
stop lists and reduce words to their stems, before creating a
frequency table
Frequency Tables
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Some Concepts
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Frequency Tables
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D is a set of N documents
T is a set of M words/terms occurring in the documents of D
Assume no words on the stop list for D occur in T and all words
in T have been stemmed
The frequency table FreqT is an (MN) matrix such that
FreqT(i,j) equals the number of occurrences of the word ti in the
document dj
Doc String
Term/Doc d1
d2
d3
d4
d1
d2
d3
d4
drug
boat
iran
connection
0
0
1
1
1
1
0
0
0
0
0
1
Sex, Drugs and Videotape
The Iranian Connection
Boating and Drugs: Slips owned by Cartel
Connections between Terrorism
and Asian Dope Operations
1
0
0
0
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Similarity
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d1 and d2 are similar because the distribution of the words
in d1 mirrors the distribution of words in d2
both contain lots of occurrences of t1 and t4 and relatively
few occurrences of t2 and t3 and moderately many
occurrences of t5
d3 and d5 are also similar
d4 and d6 stand out as sharply different
Term/Doc d1
d2
t1
t2
t3
t4
t5
390 10 10 18
4
76 217 91
8
815 142 765
511 677 11 711
33 516 64 491
615
15
2
312
45
d3
d4 d5 d6
65
816
1
2
59
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Similarity
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Is merely counting words enough?
It does not indicate the importance of the words
What about document lengths?
We should also include the importance of the word in the
document - How?
If a word occurs 3 times in a 100 word document may have
more significance than if it occurs 3 times in a million
word document
ratio of the number of occurrences of a word to the total
number of words
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Queries
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User wants to execute the query
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Find the 25 documents that are maximally relevant wrt banking
operations and drugs?
After stemming, relevant keywords are “drug, bank”
Assume the query Q as vector
We want to find the columns in FreqT that are as close as
possible to the Q’s vector
Closeness Metrics
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Term Distance: (between Q and dr) =   M j = 1 (vecQ(j) - FreqT(j,r))2
Cosine Distance: M j = 1 (vecQ(j)  FreqT(j,r))
  M j = 1 (vecQ(j))2   M j = 1 (FreqT(j,r))2
Complexity of retrievals may be O(N M) which could be very
large (Latent Semantic Indexing- A solution!!!)
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Latent Semantic Indexing
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The number of documents M and the number of terms N is
very large
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N could be over 10,000,000 (English words, proper nouns)
LSI tries to find a relatively small subset of K words which
discriminate between M documents in the archive
LSI is claimed to work effectively for around K = 200
Advantage: Each document is now a column vector of
length 200, instead of length N (This is a big plus!!!)
But, how do we find such a subset K?
A technique called singular valued decomposition
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LSI
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4 steps approach used by LSI
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Table creation: Creation of the frequency matrix FreqT
SVD Construction: Compute the singular valued decompositions
(A,S,B) of FreqT
Vector Identification: For each document d, let vec(d) be the set of
all terms in FreqT whose corresponding rows have not been
eliminated in the singular matrix S
Index Creation: Store the set of all vec(d)’s indexed by any one of
the number of techniques (such as TV-tree)
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Singular Valued Decomposition
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Let M1 and M2 are two matrices of order (m1n1) and
(m2n2), respectively
M1  M2 is well defined iff n1 = m2
3 2
4 8
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
1 4 3
2 4 6
=
7 20 21
20 48 60
Transpose of M, MT
T
7 20 21
20 48 60 =

7 20
20 48
21 60
Vector = matrix of order (1m)
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Singular Valued Decomposition
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Two vectors X and Y of the same order are said to be
orthogonal iff XTY = 0
X = [10, 5, 20], Y = [1, 2, -1]
10
0
XTY = 5  [1 2 -1] = 0
20
0
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A Matrix M is orthogonal iff MTM is the identity matrix
1 1
M =
is orthogonal
0 0
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Singular Valued Decomposition
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Matrix M is said to be diagonal iff the order of M is (mm)
and for all 1 i, j  m, i  j  M(i,j) = 0
1 0 0
1 0 0
1 1
A= 0 4 0 ; B= 0 0 0 ;C=
0 0 5
0 0 0
0 0
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A and B are diagonal, but C is not
A diagonal matrix M of order (mm) is said to be nondecreasing iff for all 1 i, j  m, i  j  M(i,i)  M(j,j)
A is a non-decreasing diagonal matrix but B is not
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SVD
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A singular value decomposition of FreqT is a triple (A,S,B)
where:
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1. FreqT = (ASBT)
2. A is an (M  M) orthogonal matrix such that ATA = I
3. B is an (N  N) orthogonal matrix such that BTB = I
4. S is a diagonal matrix called a singular matrix
Theorem: Given any matrix M of order (m  m), it is
possible to find a singular value decomposition, (A,S,B) of
M such that S is a non-decreasing diagonal matrix
The SVD of the matrix
1.44 0.52
is given by:
0.92 1.44
.6 -.8
.8 .6
5 0
.8 .6
0 2
here the singular values are 5,2
.6 -.8 and the singular matrix S is non-decreasing
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Returning to LSI
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Given a frequency matrix FreqT, we can decompose it into
SVD TSDT where S is non-decreasing
If FreqT is of size (M  N), then T is of size (M  M) and S
is of order (M  R) where R is the rank of FreqT, and DT is
of the order (R  N)
We can now shrink the problem substantially by
eliminating the least significant singular values from the
singular matrix S
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Choose an integer k that is substantially smaller than R
Replace S by S*, which is a (k  k) matrix such that S*(i,j) = S(i,j)
for 1 i, j  k
Replace the (R  N) matrix DT by the (k  N) matrix D*T where
D*T(i,j) = DT(i,j) if 1 i  k and 1 j  N
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LSI
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How?
20 0 0
0 16 0
0 0 12
0 0 .08
0 0 0
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0 0
0 0
0 0
0 0
.004
20 0 0
0 16 0
0 0 12
Bottom line:
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Throw away the least significant values and retain the rest of the
matrix
Key claim in LSI is that if k is chosen judiciously, then the k rows
appearing in the singular matrix S* represent the k “most
important” (from the point of view of retrieval) terms occurring in
the “entire” document
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Analysis
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Usually R is taken to be 200
The size of FreqT is (M  N),
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After shrinking the singular matrix to 200
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where M = number of terms = 1,000,000
N = number of documents = 10,000 (even for a small database)
the first matrix: (M  R) = 1,000,000  200 = 200,000,000
the singular matrix: (R  R) = 200  200 = 400,000 (only 200
need to be stored because all others are 0’s)
the last matrix: (R  N) = 200  10,000
A total of 202,000,200 (200 million)
In contrast, (M  N) is close to 10,000 million!!!
SVD reduced the space utilized to about 1/50th of that
required by the original frequency table
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LSI: Document Retrieval using SVD
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Given 2 documents d1 and d2 in the archive, how similar
are they?
Given a query string/document Q, what are the n
documents in the archive that are most relevant for the
query?
Dot Product
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Suppose x = (x1, … xw) and y = (y1, …, yw)
The dot product of x and y = x y = xi  yi (where i = 1,..w)
Similarity of these two documents wrt the SVD
representation TS*  D*T of a freq table is the dot product
of the two columns in the matrix D*T of the two documents
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LSI: Document Retrieval using SVD
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The top p matches for Q
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1. For all 1 i  j  p, the similarity between vecQ and di is
greater than or equal to the similarity between vecQ and dj
2. There is no other document dz such that the similarity between
dz and vecQ exceeds that of dp
Can be done by using any indexing structure for Rdimensional spaces (R-trees, k-d trees)
However R-trees, k-d trees do not work well for highdimensional data (>20)
Solution: TV-trees!
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Telescopic Vector (TV) Trees
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Access to point data in very large dimensional spaces
should be highly efficient
A document d may be viewed as a vector v of length k,
where the singular matrix is of size (k  k)
Thus each document is a point in a k-dimensional space
A document database is a collection of such points
To find the top p matches for Q, expressed as vecQ of
length k, we need to find the k-nearest neighbors vecQ
TV-tree is a data structure similar to R-trees
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Organization of a TV-tree
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NumChild: Max number of a node is allowed to have
: is a number,  > 0,  < k is the number of active
dimensions
Each in TV(k,NumChild,) represents a region, for this
purpose, each node contains 3 fields
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N.Center: this is a point in k-dimensional space
N.Radius: A real number > 0
N.ActiveDims: A list of at most  dimensions, It is a subset of
{1,…k} of cardinality  or less
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Region associated with a node N
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Suppose x and y are points in k-dim space
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Let k = 200,  = 5 and the set of ActiveDims = {1,2,3,4,5}
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x = (10,5,11,13,7,x6, ….x200)
y = (2,4,14,8,6,y6, …y200)
act-dist(x,y) = (10-2)2 + (5-4)2 + (11-14)2 + (13-8)2 + (7-6)2 = 10
Node N represents the region containing all points x such
that the active distance between x and N.Center  N.Radius
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act-dist(x,y) =   (xi - yi)2 where i ActiveDims
if N.Center = (10,5,11,13,7,0,0,0…0)
N.ActiveDims = {1,2,3,4,5}
then N represents the region consisting of all points x such that
(x1-10)2 + (x2-5)2 + (x3-14)2 + (x4-13)2 + (x5-7)2  N.Radius
A node also contains an array, Child, of pointers to other
nodes
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Properties of TV- Trees
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All data is stored at the leaf nodes
Each node (except the root and the leaves) must be at least
half full
If N is a node, and N1, .. Nr are its children, then
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Region(N) is Union of all Region(Ni)’s
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Insertion into TV-trees
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Three steps:
1. Branch Selection: When we insert a new vector v at node
N,
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for each child Nj of N, compute exp(v) = the amount we must
expand Nj.Radius so that v’s active distance from Nj.Center falls
within this region
select a branch such that exp(v) is minimum
2. Splitting: When a leaf node is full and cannot
accommodate the new vector v, we have to split.
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Split vectors into 2 groups G1,H1 such that we enclose all vectors
in G1 with center c1 and radius r1, and all in H1 with center c1’
and r1’
There exist many such cases: G2,H2 (with (c2,r2), (c2’,r2’)
take the one with minimum sum of radii, i.e., G1,H1 is better if
(r1+r1’) < (r2+r2’)
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Insertion into TV-trees
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3. Telescoping: The active dimensions associated with a
node or the children of the node change (either expand or
contract); this is called telescoping. This happens in 2
cases:
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When a node splits into two subnodes N1 and N2, vectors in
region(N1) all agree on not just the active dimensions of N, but a
few more as well
When a new vector is added to a node N, the active dimensions
may reduce
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Other Retrieval Techniques: Inverted
Indices
A document_record contains 2 fields: doc_id, postings_list
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A term_record consists of 2 fields: term, postings_list
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postings_list is list specifying which documents the term appeared
in
Two hash tables are maintained: DocTable, TermTable
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postings_list is a list of terms (or pointers to terms) that occur in
the document. Sorted using a suitable relevance measure
DocTable is constructed by hashing on doc_id
TermTable by hashing on term
To find all documents associated with a term, merely return
the postings_list
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Other Retrieval Techniques: Signature
Files
Associate a signature with each document
signature: is a representation of an ordered list of terms that
describe the document
the list of terms in the signature may be derived from a
frequency analysis, stemming, usage of stop lists
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