Latent semantic indexing Dimensionality reduction LSI in information retrieval Clustering
PV211: Introduction to Information Retrieval
https://www.fi.muni.cz/~sojka/PV211
IIR 18: Latent Semantic Indexing
Handout version
Petr Sojka, Hinrich Schütze et al.
Faculty of Informatics, Masaryk University, Brno
Center for Information and Language Processing, University of Munich
2024-03-27
(compiled on 2024-06-10 14:16:42)
Sojka, IIR Group: PV211: Latent Semantic Indexing 1 / 38
Latent semantic indexing Dimensionality reduction LSI in information retrieval Clustering
Overview
1 Latent semantic indexing
2 Dimensionality reduction
3 LSI in information retrieval
4 Clustering
Sojka, IIR Group: PV211: Latent Semantic Indexing 2 / 38
Latent semantic indexing Dimensionality reduction LSI in information retrieval Clustering
Take-away today
Latent Semantic Indexing (LSI) / Singular Value
Decomposition: The math
SVD used for dimensionality reduction
LSI: SVD in information retrieval
LSI as clustering
gensim: Topic modelling for humans (practical use of LSI et
al.)
Sojka, IIR Group: PV211: Latent Semantic Indexing 3 / 38
Latent semantic indexing Dimensionality reduction LSI in information retrieval Clustering
Recall: Term-document matrix
Anthony Julius The Hamlet Othello Macbeth
and Caesar Tempest
Cleopatra
anthony 5.25 3.18 0.0 0.0 0.0 0.35
brutus 1.21 6.10 0.0 1.0 0.0 0.0
caesar 8.59 2.54 0.0 1.51 0.25 0.0
calpurnia 0.0 1.54 0.0 0.0 0.0 0.0
cleopatra 2.85 0.0 0.0 0.0 0.0 0.0
mercy 1.51 0.0 1.90 0.12 5.25 0.88
worser 1.37 0.0 0.11 4.15 0.25 1.95
. . .
This matrix is the basis for computing the similarity between
documents and queries.
Today: Can we transform this matrix, so that we get a better
measure of similarity between documents and queries?
Sojka, IIR Group: PV211: Latent Semantic Indexing 5 / 38
Latent semantic indexing Dimensionality reduction LSI in information retrieval Clustering
Latent semantic indexing: Overview
We will decompose the term-document matrix into a product
of matrices.
The particular decomposition we’ll use: singular value
decomposition (SVD).
SVD: C = UΣV T (where C = term-document matrix)
We will then use the SVD to compute a new, improved
term-document matrix C′.
We’ll get better similarity values out of C′ (compared to C).
Using SVD for this purpose is called latent semantic indexing
or LSI.
Sojka, IIR Group: PV211: Latent Semantic Indexing 6 / 38
Latent semantic indexing Dimensionality reduction LSI in information retrieval Clustering
Example of C = UΣV T
: The matrix C
C d1 d2 d3 d4 d5 d6
ship 1 0 1 0 0 0
boat 0 1 0 0 0 0
ocean 1 1 0 0 0 0
wood 1 0 0 1 1 0
tree 0 0 0 1 0 1
This is a standard term-document matrix.
Actually, we use a non-weighted matrix here to simplify the
example.
Sojka, IIR Group: PV211: Latent Semantic Indexing 7 / 38
Latent semantic indexing Dimensionality reduction LSI in information retrieval Clustering
Example of C = UΣV T
: The matrix U
U 1 2 3 4 5
ship −0.44 −0.30 0.57 0.58 0.25
boat −0.13 −0.33 −0.59 0.00 0.73
ocean −0.48 −0.51 −0.37 0.00 −0.61
wood −0.70 0.35 0.15 −0.58 0.16
tree −0.26 0.65 −0.41 0.58 −0.09
One row per term, one column per min(M, N) where M is the
number of terms and N is the number of documents.
This is an orthonormal matrix: (i) Row vectors have unit length.
(ii) Any two distinct row vectors are orthogonal to each other.
Think of the dimensions as “semantic” dimensions that capture
distinct topics like politics, sports, economics. 2 = land/water
Each number uij in the matrix indicates how strongly related term
i is to the topic represented by semantic dimension j.
Sojka, IIR Group: PV211: Latent Semantic Indexing 8 / 38
Latent semantic indexing Dimensionality reduction LSI in information retrieval Clustering
Example of C = UΣV T
: The matrix Σ
Σ 1 2 3 4 5
1 2.16 0.00 0.00 0.00 0.00
2 0.00 1.59 0.00 0.00 0.00
3 0.00 0.00 1.28 0.00 0.00
4 0.00 0.00 0.00 1.00 0.00
5 0.00 0.00 0.00 0.00 0.39
This is a square, diagonal matrix of dimensionality
min(M, N) × min(M, N).
The diagonal consists of the singular values of C.
The magnitude of the singular value measures the importance of
the corresponding semantic dimension.
We’ll make use of this by omitting unimportant dimensions.
Sojka, IIR Group: PV211: Latent Semantic Indexing 9 / 38
Latent semantic indexing Dimensionality reduction LSI in information retrieval Clustering
Example of C = UΣV T
: The matrix V T
V T d1 d2 d3 d4 d5 d6
1 −0.75 −0.28 −0.20 −0.45 −0.33 −0.12
2 −0.29 −0.53 −0.19 0.63 0.22 0.41
3 0.28 −0.75 0.45 −0.20 0.12 −0.33
4 0.00 0.00 0.58 0.00 −0.58 0.58
5 −0.53 0.29 0.63 0.19 0.41 −0.22
One column per document, one row per min(M, N) where M is the
number of terms and N is the number of documents.
Again: This is an orthonormal matrix: (i) Column vectors have
unit length. (ii) Any two distinct column vectors are orthogonal to
each other.
These are again the semantic dimensions from matrices U and Σ
that capture distinct topics like politics, sports, economics.
Each number vij in the matrix indicates how strongly related
document i is to the topic represented by semantic dimension
j.Sojka, IIR Group: PV211: Latent Semantic Indexing 10 / 38
Latent semantic indexing Dimensionality reduction LSI in information retrieval Clustering
Example of C = UΣV T
: All four matrices Recall
unreduced decomposition C = UΣV T
Exercise: Why can
this be viewed as soft clustering?
C d1 d2 d3 d4 d5 d6
ship 1 0 1 0 0 0
boat 0 1 0 0 0 0
ocean 1 1 0 0 0 0
wood 1 0 0 1 1 0
tree 0 0 0 1 0 1
=
U 1 2 3 4 5
ship −0.44 −0.30 0.57 0.58 0.25
boat −0.13 −0.33 −0.59 0.00 0.73
ocean −0.48 −0.51 −0.37 0.00 −0.61
wood −0.70 0.35 0.15 −0.58 0.16
tree −0.26 0.65 −0.41 0.58 −0.09
×
Σ 1 2 3 4 5
1 2.16 0.00 0.00 0.00 0.00
2 0.00 1.59 0.00 0.00 0.00
3 0.00 0.00 1.28 0.00 0.00
4 0.00 0.00 0.00 1.00 0.00
5 0.00 0.00 0.00 0.00 0.39
×
V T d1 d2 d3 d4 d5 d6
1 −0.75 −0.28 −0.20 −0.45 −0.33 −0.12
2 −0.29 −0.53 −0.19 0.63 0.22 0.41
3 0.28 −0.75 0.45 −0.20 0.12 −0.33
4 0.00 0.00 0.58 0.00 −0.58 0.58
5 −0.53 0.29 0.63 0.19 0.41 −0.22
LSI is decomposition of C into a representation of the terms, a representation of the documents
and a representation of the importance of the “semantic” dimensions.
Sojka, IIR Group: PV211: Latent Semantic Indexing 11 / 38
Latent semantic indexing Dimensionality reduction LSI in information retrieval Clustering
LSI: Summary
We’ve decomposed the term-document matrix C into a
product of three matrices: UΣV T .
The term matrix U – consists of one (row) vector for each
term
The document matrix V T – consists of one (column) vector
for each document
The singular value matrix Σ – diagonal matrix with singular
values, reflecting importance of each dimension
Next: Why are we doing this?
Sojka, IIR Group: PV211: Latent Semantic Indexing 12 / 38
Latent semantic indexing Dimensionality reduction LSI in information retrieval Clustering
Exercise
V T d1 d2 d3 d4 d5 d6
1 −0.75 −0.28 −0.20 −0.45 −0.33 −0.12
2 −0.29 −0.53 −0.19 0.63 0.22 0.41
3 0.28 −0.75 0.45 −0.20 0.12 −0.33
4 0.00 0.00 0.58 0.00 −0.58 0.58
5 −0.53 0.29 0.63 0.19 0.41 −0.22
Verify that the first document has unit length.
Verify that the first two documents are orthogonal.
0.752 + 0.292 + 0.282 + 0.002 + 0.532 = 1.0059
−0.75 ∗ −0.28 + −0.29 ∗ −0.53 + 0.28 ∗ −0.75 + 0.00 ∗ 0.00 +
−0.53 ∗ 0.29 = 0
Sojka, IIR Group: PV211: Latent Semantic Indexing 13 / 38
Latent semantic indexing Dimensionality reduction LSI in information retrieval Clustering
How we use the SVD in LSI
Key property: Each singular value tells us how important its
dimension is.
By setting less important dimensions to zero, we keep the
important information, but get rid of the “details”.
These details may
be noise – in that case, reduced LSI is a better representation
because it is less noisy.
make things dissimilar that should be similar – again, the
reduced LSI representation is a better representation because it
represents similarity better.
Analogy for “fewer details is better”
Image of a blue flower
Image of a yellow flower
Omitting color makes it easier to see the similarity
Sojka, IIR Group: PV211: Latent Semantic Indexing 15 / 38
Latent semantic indexing Dimensionality reduction LSI in information retrieval Clustering
Reducing the dimensionality to 2
U 1 2 3 4 5
ship −0.44 −0.30 0.00 0.00 0.00
boat −0.13 −0.33 0.00 0.00 0.00
ocean −0.48 −0.51 0.00 0.00 0.00
wood −0.70 0.35 0.00 0.00 0.00
tree −0.26 0.65 0.00 0.00 0.00
Σ2 1 2 3 4 5
1 2.16 0.00 0.00 0.00 0.00
2 0.00 1.59 0.00 0.00 0.00
3 0.00 0.00 0.00 0.00 0.00
4 0.00 0.00 0.00 0.00 0.00
5 0.00 0.00 0.00 0.00 0.00
V T d1 d2 d3 d4 d5 d6
1 −0.75 −0.28 −0.20 −0.45 −0.33 −0.12
2 −0.29 −0.53 −0.19 0.63 0.22 0.41
3 0.00 0.00 0.00 0.00 0.00 0.00
4 0.00 0.00 0.00 0.00 0.00 0.00
5 0.00 0.00 0.00 0.00 0.00 0.00
Actually, we
only zero out
singular values
in Σ. This has
the effect of
setting the
corresponding
dimensions in
U and V T
to
zero when
computing the
product C =
UΣV T
.
Sojka, IIR Group: PV211: Latent Semantic Indexing 16 / 38
Latent semantic indexing Dimensionality reduction LSI in information retrieval Clustering
Reducing the dimensionality to 2
C2 d1 d2 d3 d4 d5 d6
ship 0.85 0.52 0.28 0.13 0.21 −0.08
boat 0.36 0.36 0.16 −0.20 −0.02 −0.18
ocean 1.01 0.72 0.36 −0.04 0.16 −0.21
wood 0.97 0.12 0.20 1.03 0.62 0.41
tree 0.12 −0.39 −0.08 0.90 0.41 0.49
=
U 1 2 3 4 5
ship −0.44 −0.30 0.57 0.58 0.25
boat −0.13 −0.33 −0.59 0.00 0.73
ocean −0.48 −0.51 −0.37 0.00 −0.61
wood −0.70 0.35 0.15 −0.58 0.16
tree −0.26 0.65 −0.41 0.58 −0.09
×
Σ2 1 2 3 4 5
1 2.16 0.00 0.00 0.00 0.00
2 0.00 1.59 0.00 0.00 0.00
3 0.00 0.00 0.00 0.00 0.00
4 0.00 0.00 0.00 0.00 0.00
5 0.00 0.00 0.00 0.00 0.00
×
V T d1 d2 d3 d4 d5 d6
1 −0.75 −0.28 −0.20 −0.45 −0.33 −0.12
2 −0.29 −0.53 −0.19 0.63 0.22 0.41
3 0.28 −0.75 0.45 −0.20 0.12 −0.33
4 0.00 0.00 0.58 0.00 −0.58 0.58
5 −0.53 0.29 0.63 0.19 0.41 −0.22
Sojka, IIR Group: PV211: Latent Semantic Indexing 17 / 38
Latent semantic indexing Dimensionality reduction LSI in information retrieval Clustering
Example of C = UΣV T
: All four matrices Recall
unreduced decomposition C = UΣV T
Exercise: Why can
this be viewed as soft clustering?
C d1 d2 d3 d4 d5 d6
ship 1 0 1 0 0 0
boat 0 1 0 0 0 0
ocean 1 1 0 0 0 0
wood 1 0 0 1 1 0
tree 0 0 0 1 0 1
=
U 1 2 3 4 5
ship −0.44 −0.30 0.57 0.58 0.25
boat −0.13 −0.33 −0.59 0.00 0.73
ocean −0.48 −0.51 −0.37 0.00 −0.61
wood −0.70 0.35 0.15 −0.58 0.16
tree −0.26 0.65 −0.41 0.58 −0.09
×
Σ 1 2 3 4 5
1 2.16 0.00 0.00 0.00 0.00
2 0.00 1.59 0.00 0.00 0.00
3 0.00 0.00 1.28 0.00 0.00
4 0.00 0.00 0.00 1.00 0.00
5 0.00 0.00 0.00 0.00 0.39
×
V T d1 d2 d3 d4 d5 d6
1 −0.75 −0.28 −0.20 −0.45 −0.33 −0.12
2 −0.29 −0.53 −0.19 0.63 0.22 0.41
3 0.28 −0.75 0.45 −0.20 0.12 −0.33
4 0.00 0.00 0.58 0.00 −0.58 0.58
5 −0.53 0.29 0.63 0.19 0.41 −0.22
LSI is decomposition of C into a representation of the terms, a representation of the documents
and a representation of the importance of the “semantic” dimensions.
Sojka, IIR Group: PV211: Latent Semantic Indexing 18 / 38
Latent semantic indexing Dimensionality reduction LSI in information retrieval Clustering
Original matrix C vs. reduced C2 = UΣ2V T
C d1 d2 d3 d4 d5 d6
ship 1 0 1 0 0 0
boat 0 1 0 0 0 0
ocean 1 1 0 0 0 0
wood 1 0 0 1 1 0
tree 0 0 0 1 0 1
C2 d1 d2 d3 d4 d5 d6
ship 0.85 0.52 0.28 0.13 0.21 −0.08
boat 0.36 0.36 0.16 −0.20 −0.02 −0.18
ocean 1.01 0.72 0.36 −0.04 0.16 −0.21
wood 0.97 0.12 0.20 1.03 0.62 0.41
tree 0.12 −0.39 −0.08 0.90 0.41 0.49
We can view
C2 as a two-
dimensional
representation
of the matrix
C. We have
performed a
dimensionality
reduction to
two
dimensions.
Sojka, IIR Group: PV211: Latent Semantic Indexing 19 / 38
Latent semantic indexing Dimensionality reduction LSI in information retrieval Clustering
Exercise
C d1 d2 d3 d4 d5 d6
ship 1 0 1 0 0 0
boat 0 1 0 0 0 0
ocean 1 1 0 0 0 0
wood 1 0 0 1 1 0
tree 0 0 0 1 0 1
C2 d1 d2 d3 d4 d5 d6
ship 0.85 0.52 0.28 0.13 0.21 −0.08
boat 0.36 0.36 0.16 −0.20 −0.02 −0.18
ocean 1.01 0.72 0.36 −0.04 0.16 −0.21
wood 0.97 0.12 0.20 1.03 0.62 0.41
tree 0.12 −0.39 −0.08 0.90 0.41 0.49
Compute the
similarity between
d2 and d3 for the
original matrix
and for the
reduced matrix.
Sojka, IIR Group: PV211: Latent Semantic Indexing 20 / 38
Latent semantic indexing Dimensionality reduction LSI in information retrieval Clustering
Why the reduced matrix C2 is better than C
C d1 d2 d3 d4 d5 d6
ship 1 0 1 0 0 0
boat 0 1 0 0 0 0
ocean 1 1 0 0 0 0
wood 1 0 0 1 1 0
tree 0 0 0 1 0 1
C2 d1 d2 d3 d4 d5 d6
ship 0.85 0.52 0.28 0.13 0.21 −0.08
boat 0.36 0.36 0.16 −0.20 −0.02 −0.18
ocean 1.01 0.72 0.36 −0.04 0.16 −0.21
wood 0.97 0.12 0.20 1.03 0.62 0.41
tree 0.12 −0.39 −0.08 0.90 0.41 0.49
Similarity of d2 and d3 in the
original space: 0.
Similarity of d2 and d3 in the
reduced space:
0.52 ∗ 0.28 + 0.36 ∗ 0.16 +
0.72 ∗ 0.36 + 0.12 ∗ 0.20 +
−0.39 ∗ −0.08 ≈ 0.52
“boat” and “ship” are
semantically similar. The
“reduced” similarity measure
reflects this.
What property of the SVD
reduction is responsible for
improved similarity?
Sojka, IIR Group: PV211: Latent Semantic Indexing 21 / 38
Latent semantic indexing Dimensionality reduction LSI in information retrieval Clustering
Exercise: Compute matrix product
C2 d1 d2 d3 d4 d5 d6
ship 0.09 0.16 0.06 -0.19 -0.07 -0.12
boat 0.10 0.17 0.06 -0.21 -0.07 -0.14
ocean 0.15 0.27 0.10 -0.32 -0.11 -0.21
wood -0.10 -0.19 -0.07 0.22 0.08 0.14
tree -0.19 -0.34 -0.12 0.41 0.14 0.27
???????=
U 1 2 3 4 5
ship −0.44 −0.30 0.57 0.58 0.25
boat −0.13 −0.33 −0.59 0.00 0.73
ocean −0.48 −0.51 −0.37 0.00 −0.61
wood −0.70 0.35 0.15 −0.58 0.16
tree −0.26 0.65 −0.41 0.58 −0.09
×
Σ2 1 2 3 4 5
1 0.00 0.00 0.00 0.00 0.00
2 0.00 1.59 0.00 0.00 0.00
3 0.00 0.00 0.00 0.00 0.00
4 0.00 0.00 0.00 0.00 0.00
5 0.00 0.00 0.00 0.00 0.00
×
V T d1 d2 d3 d4 d5 d6
1 −0.75 −0.28 −0.20 −0.45 −0.33 −0.12
2 −0.29 −0.53 −0.19 0.63 0.22 0.41
3 0.28 −0.75 0.45 −0.20 0.12 −0.33
4 0.00 0.00 0.58 0.00 −0.58 0.58
5 −0.53 0.29 0.63 0.19 0.41 −0.22
Sojka, IIR Group: PV211: Latent Semantic Indexing 22 / 38
Latent semantic indexing Dimensionality reduction LSI in information retrieval Clustering
Why we use LSI in information retrieval
LSI takes documents that are semantically similar (= talk
about the same topics), . . .
. . . but are not similar in the vector space (because they use
different words) . . .
. . . and re-represents them in a reduced vector space . . .
. . . in which they have higher similarity.
Thus, LSI addresses the problems of synonymy and semantic
relatedness.
Standard vector space: Synonyms contribute nothing to
document similarity.
Desired effect of LSI: Synonyms contribute strongly to
document similarity.
Sojka, IIR Group: PV211: Latent Semantic Indexing 24 / 38
Latent semantic indexing Dimensionality reduction LSI in information retrieval Clustering
How LSI addresses synonymy and semantic relatedness
The dimensionality reduction forces us to omit a lot of
“detail”.
We have to map differents words (= different dimensions of
the full space) to the same dimension in the reduced space.
The “cost” of mapping synonyms to the same dimension is
much less than the cost of collapsing unrelated words.
SVD selects the “least costly” mapping (see below).
Thus, it will map synonyms to the same dimension.
But it will avoid doing that for unrelated words.
Sojka, IIR Group: PV211: Latent Semantic Indexing 25 / 38
Latent semantic indexing Dimensionality reduction LSI in information retrieval Clustering
LSI: Comparison to other approaches
Recap: Relevance feedback and query expansion are used to
increase recall in information retrieval – if query and
documents have no terms in common.
(or, more commonly, too few terms in common for a high
similarity score)
LSI increases recall and hurts precision.
Thus, it addresses the same problems as (pseudo) relevance
feedback and query expansion . . .
. . . and it has the same problems.
Sojka, IIR Group: PV211: Latent Semantic Indexing 26 / 38
Latent semantic indexing Dimensionality reduction LSI in information retrieval Clustering
Implementation
Compute SVD of term-document matrix
Reduce the space and compute reduced document
representations
Map the query into the reduced space qk = Σ−1
k UT
k q.
This follows from: Ck = UkΣkV T
k ⇒ Σ−1
k UT C = V T
k
Compute similarity of qk with all reduced documents in Vk.
Output ranked list of documents as usual
Exercise: What is the fundamental problem with this
approach?
Sojka, IIR Group: PV211: Latent Semantic Indexing 27 / 38
Latent semantic indexing Dimensionality reduction LSI in information retrieval Clustering
Optimality
SVD is optimal in the following sense.
Keeping the k largest singular values and setting all others to
zero gives you the optimal approximation of the original
matrix C. Eckart-Young theorem
Optimal: no other matrix of the same rank (= with the same
underlying dimensionality) approximates C better.
Measure of approximation is Frobenius norm:
||C||F = i j c2
ij
So LSI uses the “best possible” matrix.
There is only one best possible matrix – unique solution
(modulo signs).
Caveat: There is only a tenuous relationship between the
Frobenius norm and cosine similarity between documents.
Sojka, IIR Group: PV211: Latent Semantic Indexing 28 / 38
Latent semantic indexing Dimensionality reduction LSI in information retrieval Clustering
Data for graphical illustration of LSI
c1 Human machine interface for lab abc computer applications
c2 A survey of user opinion of computer system response time
c3 The EPS user interface management system
c4 System and human system engineering testing of EPS
c5 Relation of user perceived response time to error measurement
m1 The generation of random binary unordered trees
m2 The intersection graph of paths in trees
m3 Graph minors IV Widths of trees and well quasi ordering
m4 Graph minors A survey
The matrix C
c1 c2 c3 c4 c5 m1 m2 m3 m4
human 1 0 0 1 0 0 0 0 0
interface 1 0 1 0 0 0 0 0 0
computer 1 1 0 0 0 0 0 0 0
user 0 1 1 0 1 0 0 0 0
system 0 1 1 2 0 0 0 0 0
response 0 1 0 0 1 0 0 0 0
time 0 1 0 0 1 0 0 0 0
EPS 0 0 1 1 0 0 0 0 0
survey 0 1 0 0 0 0 0 0 1
trees 0 0 0 0 0 1 1 1 0
graph 0 0 0 0 0 0 1 1 1
minors 0 0 0 0 0 0 0 1 1
Sojka, IIR Group: PV211: Latent Semantic Indexing 29 / 38
Latent semantic indexing Dimensionality reduction LSI in information retrieval Clustering
Graphical illustration of LSI: Plot of C2
2-dimensional plot of
C2 (scaled dimensions).
Circles = terms. Open
squares = documents
(component terms in
parentheses). q = query
“human computer inter-
action”.
The dotted cone represents the region whose points are within a cosine of
.9 from q . All documents about human-computer documents (c1-c5) are
near q, even c3/c5 although they share no terms. None of the graph theory
documents (m1-m4) are near q.
Sojka, IIR Group: PV211: Latent Semantic Indexing 30 / 38
Latent semantic indexing Dimensionality reduction LSI in information retrieval Clustering
Exercise
What happens when we rank documents according to cosine
similarity in the original vector space? What happens when we
rank documents according to cosine similarity in the reduced vector
space?
Sojka, IIR Group: PV211: Latent Semantic Indexing 31 / 38
Latent semantic indexing Dimensionality reduction LSI in information retrieval Clustering
LSI performs better than vector space on MED collection
LSI-100 = LSI reduced to 100 dimensions; SMART = SMART
implementation of vector space model
Sojka, IIR Group: PV211: Latent Semantic Indexing 32 / 38
Latent semantic indexing Dimensionality reduction LSI in information retrieval Clustering
Example of C = UΣV T
: All four matrices Recall
unreduced decomposition C = UΣV T
Exercise: Why can
this be viewed as soft clustering?
C d1 d2 d3 d4 d5 d6
ship 1 0 1 0 0 0
boat 0 1 0 0 0 0
ocean 1 1 0 0 0 0
wood 1 0 0 1 1 0
tree 0 0 0 1 0 1
=
U 1 2 3 4 5
ship −0.44 −0.30 0.57 0.58 0.25
boat −0.13 −0.33 −0.59 0.00 0.73
ocean −0.48 −0.51 −0.37 0.00 −0.61
wood −0.70 0.35 0.15 −0.58 0.16
tree −0.26 0.65 −0.41 0.58 −0.09
×
Σ 1 2 3 4 5
1 2.16 0.00 0.00 0.00 0.00
2 0.00 1.59 0.00 0.00 0.00
3 0.00 0.00 1.28 0.00 0.00
4 0.00 0.00 0.00 1.00 0.00
5 0.00 0.00 0.00 0.00 0.39
×
V T d1 d2 d3 d4 d5 d6
1 −0.75 −0.28 −0.20 −0.45 −0.33 −0.12
2 −0.29 −0.53 −0.19 0.63 0.22 0.41
3 0.28 −0.75 0.45 −0.20 0.12 −0.33
4 0.00 0.00 0.58 0.00 −0.58 0.58
5 −0.53 0.29 0.63 0.19 0.41 −0.22
LSI is decomposition of C into a representation of the terms, a representation of the documents
and a representation of the importance of the “semantic” dimensions.
Sojka, IIR Group: PV211: Latent Semantic Indexing 34 / 38
Latent semantic indexing Dimensionality reduction LSI in information retrieval Clustering
Why LSI can be viewed as soft clustering
Each of the k dimensions of the reduced space is one cluster.
If the value of the LSI representation of document d on
dimension k is x, then x is the soft membership of d in
topic k.
This soft membership can be positive or negative.
Example: Dimension 2 in our SVD decomposition
This dimension/cluster corresponds to the water/earth
dichotomy.
“ship”, “boat”, “ocean” have negative values.
“wood”, “tree” have positive values.
d1, d2, d3 have negative values (most of their terms are water
terms).
d4, d5, d6 have positive values (all of their terms are earth
terms).
Sojka, IIR Group: PV211: Latent Semantic Indexing 35 / 38
Latent semantic indexing Dimensionality reduction LSI in information retrieval Clustering
Semantic indexing and clustering with Gensim
Gensim: an open-source vector space modeling and topic modeling
toolkit, implemented in the Python programming language
Tutorial examples of topic modelling for humans (LSI):
http://radimrehurek.com/gensim/tut2.html
DML-CZ similarity example:
http://dml.cz/handle/10338.dmlcz/500114/SimilarArticles
cf. papers similar to famous Otakar Borůvka’s paper
Go forth and create masterpieces for semantic indexing
applications (by gensim, similarly as other 5300+ already did ;-)!
Sojka, IIR Group: PV211: Latent Semantic Indexing 36 / 38
Latent semantic indexing Dimensionality reduction LSI in information retrieval Clustering
Take-away today
Latent Semantic Indexing (LSI) / Singular Value
Decomposition: The math
SVD used for dimensionality reduction
LSI: SVD in information retrieval
LSI as clustering
gensim: Topic modelling for humans (practical use of LSI et
al.)
Sojka, IIR Group: PV211: Latent Semantic Indexing 37 / 38
Latent semantic indexing Dimensionality reduction LSI in information retrieval Clustering
Resources
Chapter 18 of IIR
Resources at https://www.fi.muni.cz/~sojka/PV211/
and http://cislmu.org, materials in MU IS and FI MU
library
Original paper on latent semantic indexing by Deerwester et al.
Paper on probabilistic LSI by Thomas Hofmann
Word space: LSI for words
Sojka, IIR Group: PV211: Latent Semantic Indexing 38 / 38