1. Text Databases Text Types
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
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

2. We discuss, Measures of performance of a text retrieval system
Latent semantic indexing
Telescopic-Vector trees for document retrieval

3. Precision and Recall Precision: Recall: 50 20 150
How many of the returned documents are relevant?
(20+1)/( )
Recall:
How many of the relevant documents are returned?
(20+1)/( ) 50
Returned documents 20
150
Relevant documents
All documents

4. Some Concepts Stop List Word Stems
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
Word Stems
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

5. Some Concepts Frequency Tables 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
d1 Sex, Drugs and Videotape
d2 The Iranian Connection
d3 Boating and Drugs: Slips owned by Cartel
d4 Connections between Terrorism
and Asian Dope Operations
Term/Doc d1 d d3 d4
drug
boat
iran
connection

6. Similarity
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 d d3 d4 d5 d6 t t t t t

7. Similarity 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

8. Queries User wants to execute the query
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
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!!!)

9. Latent Semantic Indexing
The number of documents M and the number of terms N is very large
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

10. LSI 4 steps approach used by LSI
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)

11. Singular Valued Decomposition
Let M1 and M2 are two matrices of order (m1n1) and (m2n2), respectively
M1  M2 is well defined iff n1 = m2
Transpose of M, MT
Vector = matrix of order (1m)
 = T =
21 60

12. Singular Valued Decomposition
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]
A Matrix M is orthogonal iff MTM is the identity matrix
XTY =  [ ] = 0
1 1
M = is orthogonal
0 0

13. Singular Valued Decomposition
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
A and B are diagonal, but C is not
A diagonal matrix M of order (mm) is said to be non-decreasing 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
A = ; B = ; C =

14. SVD A singular value decomposition of FreqT is a triple (A,S,B) where:
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
is given by:
here the singular values are 5,2
and the singular matrix S is non-decreasing

15. Returning to LSI
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
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

16. LSI
How?
Bottom line:
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

17. Analysis Usually R is taken to be 200 The size of FreqT is (M  N),
where M = number of terms = 1,000,000
N = number of documents = 10,000 (even for a small database)
After shrinking the singular matrix to 200
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

18. LSI: Document Retrieval using SVD
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
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

19. LSI: Document Retrieval using SVD
The top p matches for Q
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 R-dimensional spaces (R-trees, k-d trees)
However R-trees, k-d trees do not work well for high-dimensional data (>20)
Solution: TV-trees!

20. Telescopic Vector (TV) Trees
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

21. Organization of a TV-tree
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
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

22. Region associated with a node N
Suppose x and y are points in k-dim space
act-dist(x,y) =   (xi - yi)2 where i ActiveDims
Let k = 200,  = 5 and the set of ActiveDims = {1,2,3,4,5}
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
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

23. Properties of TV- Trees
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
Region(N) is Union of all Region(Ni)’s

24. Insertion into TV-trees
Three steps:
1. Branch Selection: When we insert a new vector v at node N,
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.
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’)

25. Insertion into TV-trees
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:
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

26. Other Retrieval Techniques: Inverted Indices
A document_record contains 2 fields: doc_id, postings_list
postings_list is a list of terms (or pointers to terms) that occur in the document. Sorted using a suitable relevance measure
A term_record consists of 2 fields: term, postings_list
postings_list is list specifying which documents the term appeared in
Two hash tables are maintained: DocTable, TermTable
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

27. 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