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CMU SCS
15-826: Multimedia Databases and Data Mining
Lecture #18: SVD - part I (definitions)
C. Faloutsos
CMU SCS
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Must-read Material
• Numerical Recipes in C ch. 2.6;
• MM Textbook Appendix D
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Outline
Goal: ‘Find similar / interesting things’
• Intro to DB
• Indexing - similarity search
• Data Mining
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Indexing - Detailed outline• primary key indexing• secondary key / multi-key indexing• spatial access methods• fractals• text• Singular Value Decomposition (SVD)• multimedia• ...
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SVD - Detailed outline• Motivation• Definition - properties• Interpretation• Complexity• Case studies• Additional properties
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SVD - Motivation• problem #1: text - LSI: find ‘concepts’• problem #2: compression / dim. reduction
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SVD - Motivation• problem #1: text - LSI: find ‘concepts’
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SVD - Motivation• Customer-product, for recommendation
system:
1 1 1 0 0
2 2 2 0 0
1 1 1 0 0
5 5 5 0 0
0 0 0 2 2
0 0 0 3 30 0 0 1 1
bread
lettu
cebe
ef
vegetarians
meat eaters
tom
atos
chick
en
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SVD - Motivation• problem #2: compress / reduce
dimensionality
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Problem - specs
• ~10**6 rows; ~10**3 columns; no updates;
• random access to any cell(s) ; small error: OK
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SVD - Motivation
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SVD - Motivation
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SVD - Detailed outline• Motivation• Definition - properties• Interpretation• Complexity• Case studies• Additional properties
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SVD - Definition(reminder: matrix multiplication
1 2
3 45 6
x 1
-1
3 x 2 2 x 1
=
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SVD - Definition(reminder: matrix multiplication
1 2
3 45 6
x 1
-1
3 x 2 2 x 1
=
3 x 1
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SVD - Definition(reminder: matrix multiplication
1 2
3 45 6
x 1
-1
3 x 2 2 x 1
=
-1
3 x 1
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SVD - Definition(reminder: matrix multiplication
1 2
3 45 6
x 1
-1
3 x 2 2 x 1
=
-1
-1
3 x 1
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SVD - Definition(reminder: matrix multiplication
1 2
3 45 6
x 1
-1=
-1
-1-1
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SVD - Definition
A[n x m] = U[n x r] r x r] (V[m x r])T
• A: n x m matrix (eg., n documents, m terms)• U: n x r matrix (n documents, r concepts)• : r x r diagonal matrix (strength of each
‘concept’) (r : rank of the matrix)• V: m x r matrix (m terms, r concepts)
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SVD - Definition• A = U VT - example:
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SVD - PropertiesTHEOREM [Press+92]: always possible to
decompose matrix A into A = U VT , where• U, V: unique (*)• U, V: column orthonormal (ie., columns are unit
vectors, orthogonal to each other)– UT U = I; VT V = I (I: identity matrix)
• : singular are positive, and sorted in decreasing order
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SVD - Example• A = U VT - example:
1 1 1 0 0
2 2 2 0 0
1 1 1 0 0
5 5 5 0 0
0 0 0 2 2
0 0 0 3 30 0 0 1 1
datainf.
retrieval
brain lung
0.18 0
0.36 0
0.18 0
0.90 0
0 0.53
0 0.800 0.27
=CS
MD
9.64 0
0 5.29x
0.58 0.58 0.58 0 0
0 0 0 0.71 0.71
x
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SVD - Example• A = U VT - example:
1 1 1 0 0
2 2 2 0 0
1 1 1 0 0
5 5 5 0 0
0 0 0 2 2
0 0 0 3 30 0 0 1 1
datainf.
retrieval
brain lung
0.18 0
0.36 0
0.18 0
0.90 0
0 0.53
0 0.800 0.27
=CS
MD
9.64 0
0 5.29x
0.58 0.58 0.58 0 0
0 0 0 0.71 0.71
x
CS-conceptMD-concept
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SVD - Example• A = U VT - example:
1 1 1 0 0
2 2 2 0 0
1 1 1 0 0
5 5 5 0 0
0 0 0 2 2
0 0 0 3 30 0 0 1 1
datainf.
retrieval
brain lung
0.18 0
0.36 0
0.18 0
0.90 0
0 0.53
0 0.800 0.27
=CS
MD
9.64 0
0 5.29x
0.58 0.58 0.58 0 0
0 0 0 0.71 0.71
x
CS-conceptMD-concept
doc-to-concept similarity matrix
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SVD - Example• A = U VT - example:
1 1 1 0 0
2 2 2 0 0
1 1 1 0 0
5 5 5 0 0
0 0 0 2 2
0 0 0 3 30 0 0 1 1
datainf.
retrieval
brain lung
0.18 0
0.36 0
0.18 0
0.90 0
0 0.53
0 0.800 0.27
=CS
MD
9.64 0
0 5.29x
0.58 0.58 0.58 0 0
0 0 0 0.71 0.71
x
‘strength’ of CS-concept
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SVD - Example• A = U VT - example:
1 1 1 0 0
2 2 2 0 0
1 1 1 0 0
5 5 5 0 0
0 0 0 2 2
0 0 0 3 30 0 0 1 1
datainf.
retrieval
brain lung
0.18 0
0.36 0
0.18 0
0.90 0
0 0.53
0 0.800 0.27
=CS
MD
9.64 0
0 5.29x
0.58 0.58 0.58 0 0
0 0 0 0.71 0.71
x
term-to-conceptsimilarity matrix
CS-concept
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SVD - Example• A = U VT - example:
1 1 1 0 0
2 2 2 0 0
1 1 1 0 0
5 5 5 0 0
0 0 0 2 2
0 0 0 3 30 0 0 1 1
datainf.
retrieval
brain lung
0.18 0
0.36 0
0.18 0
0.90 0
0 0.53
0 0.800 0.27
=CS
MD
9.64 0
0 5.29x
0.58 0.58 0.58 0 0
0 0 0 0.71 0.71
x
term-to-conceptsimilarity matrix
CS-concept
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SVD - Detailed outline• Motivation• Definition - properties• Interpretation• Complexity• Case studies• Additional properties
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SVD - Interpretation #1‘documents’, ‘terms’ and ‘concepts’:• U: document-to-concept similarity matrix• V: term-to-concept sim. matrix: its diagonal elements: ‘strength’ of each
concept
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SVD – Interpretation #1‘documents’, ‘terms’ and ‘concepts’:Q: if A is the document-to-term matrix, what
is AT A?A:Q: A AT ?A:
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SVD – Interpretation #1‘documents’, ‘terms’ and ‘concepts’:Q: if A is the document-to-term matrix, what
is AT A?A: term-to-term ([m x m]) similarity matrixQ: A AT ?A: document-to-document ([n x n]) similarity
matrix
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SVD properties• V are the eigenvectors of the covariance
matrix ATA
• U are the eigenvectors of the Gram (inner-product) matrix AAT
Further reading:1. Ian T. Jolliffe, Principal Component Analysis (2nd ed), Springer, 2002.2. Gilbert Strang, Linear Algebra and Its Applications (4th ed), Brooks Cole, 2005.
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SVD - Interpretation #2• best axis to project on: (‘best’ = min sum of
squares of projection errors)
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SVD - Motivation
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SVD - interpretation #2
• minimum RMS error
SVD: givesbest axis to project
v1
first singular
vector
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SVD - Interpretation #2
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SVD - Interpretation #2• A = U VT - example:
1 1 1 0 0
2 2 2 0 0
1 1 1 0 0
5 5 5 0 0
0 0 0 2 2
0 0 0 3 30 0 0 1 1
0.18 0
0.36 0
0.18 0
0.90 0
0 0.53
0 0.800 0.27
=9.64 0
0 5.29x
0.58 0.58 0.58 0 0
0 0 0 0.71 0.71
x
v1
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SVD - Interpretation #2• A = U VT - example:
1 1 1 0 0
2 2 2 0 0
1 1 1 0 0
5 5 5 0 0
0 0 0 2 2
0 0 0 3 30 0 0 1 1
0.18 0
0.36 0
0.18 0
0.90 0
0 0.53
0 0.800 0.27
=9.64 0
0 5.29x
0.58 0.58 0.58 0 0
0 0 0 0.71 0.71
x
variance (‘spread’) on the v1 axis
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SVD - Interpretation #2• A = U VT - example:
– U gives the coordinates of the points in the projection axis
1 1 1 0 0
2 2 2 0 0
1 1 1 0 0
5 5 5 0 0
0 0 0 2 2
0 0 0 3 30 0 0 1 1
0.18 0
0.36 0
0.18 0
0.90 0
0 0.53
0 0.800 0.27
=9.64 0
0 5.29x
0.58 0.58 0.58 0 0
0 0 0 0.71 0.71
x
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SVD - Interpretation #2• More details• Q: how exactly is dim. reduction done?
1 1 1 0 0
2 2 2 0 0
1 1 1 0 0
5 5 5 0 0
0 0 0 2 2
0 0 0 3 30 0 0 1 1
0.18 0
0.36 0
0.18 0
0.90 0
0 0.53
0 0.800 0.27
=9.64 0
0 5.29x
0.58 0.58 0.58 0 0
0 0 0 0.71 0.71
x
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SVD - Interpretation #2• More details• Q: how exactly is dim. reduction done?• A: set the smallest singular values to zero:
1 1 1 0 0
2 2 2 0 0
1 1 1 0 0
5 5 5 0 0
0 0 0 2 2
0 0 0 3 30 0 0 1 1
0.18 0
0.36 0
0.18 0
0.90 0
0 0.53
0 0.800 0.27
=9.64 0
0 5.29x
0.58 0.58 0.58 0 0
0 0 0 0.71 0.71
x
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SVD - Interpretation #2
1 1 1 0 0
2 2 2 0 0
1 1 1 0 0
5 5 5 0 0
0 0 0 2 2
0 0 0 3 30 0 0 1 1
0.18 0
0.36 0
0.18 0
0.90 0
0 0.53
0 0.800 0.27
~9.64 0
0 0x
0.58 0.58 0.58 0 0
0 0 0 0.71 0.71
x
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SVD - Interpretation #2
1 1 1 0 0
2 2 2 0 0
1 1 1 0 0
5 5 5 0 0
0 0 0 2 2
0 0 0 3 30 0 0 1 1
0.18 0
0.36 0
0.18 0
0.90 0
0 0.53
0 0.800 0.27
~9.64 0
0 0x
0.58 0.58 0.58 0 0
0 0 0 0.71 0.71
x
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SVD - Interpretation #2
1 1 1 0 0
2 2 2 0 0
1 1 1 0 0
5 5 5 0 0
0 0 0 2 2
0 0 0 3 30 0 0 1 1
0.18
0.36
0.18
0.90
0
00
~9.64
x
0.58 0.58 0.58 0 0
x
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SVD - Interpretation #2
1 1 1 0 0
2 2 2 0 0
1 1 1 0 0
5 5 5 0 0
0 0 0 2 2
0 0 0 3 30 0 0 1 1
~
1 1 1 0 0
2 2 2 0 0
1 1 1 0 0
5 5 5 0 0
0 0 0 0 0
0 0 0 0 00 0 0 0 0
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SVD - Interpretation #2Exactly equivalent:‘spectral decomposition’ of the matrix:
1 1 1 0 0
2 2 2 0 0
1 1 1 0 0
5 5 5 0 0
0 0 0 2 2
0 0 0 3 30 0 0 1 1
0.18 0
0.36 0
0.18 0
0.90 0
0 0.53
0 0.800 0.27
=9.64 0
0 5.29x
0.58 0.58 0.58 0 0
0 0 0 0.71 0.71
x
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SVD - Interpretation #2Exactly equivalent:‘spectral decomposition’ of the matrix:
1 1 1 0 0
2 2 2 0 0
1 1 1 0 0
5 5 5 0 0
0 0 0 2 2
0 0 0 3 30 0 0 1 1
= x xu1 u2
1
2
v1
v2
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SVD - Interpretation #2Exactly equivalent:‘spectral decomposition’ of the matrix:
1 1 1 0 0
2 2 2 0 0
1 1 1 0 0
5 5 5 0 0
0 0 0 2 2
0 0 0 3 30 0 0 1 1
= u11 vT1 u22 vT
2+ +...n
m
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SVD - Interpretation #2Exactly equivalent:‘spectral decomposition’ of the matrix:
1 1 1 0 0
2 2 2 0 0
1 1 1 0 0
5 5 5 0 0
0 0 0 2 2
0 0 0 3 30 0 0 1 1
= u11 vT1 u22 vT
2+ +...n
m
n x 1 1 x m
r terms
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SVD - Interpretation #2approximation / dim. reduction:by keeping the first few terms (Q: how many?)
1 1 1 0 0
2 2 2 0 0
1 1 1 0 0
5 5 5 0 0
0 0 0 2 2
0 0 0 3 30 0 0 1 1
= u11 vT1 u22 vT
2+ +...n
m
assume: 1 >= 2 >= ...
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SVD - Interpretation #2A (heuristic - [Fukunaga]): keep 80-90% of
‘energy’ (= sum of squares of i ’s)
1 1 1 0 0
2 2 2 0 0
1 1 1 0 0
5 5 5 0 0
0 0 0 2 2
0 0 0 3 30 0 0 1 1
= u11 vT1 u22 vT
2+ +...n
m
assume: 1 >= 2 >= ...
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Pictorially: matrix form of SVD
– Best rank-k approximation in L2
Am
n
m
n
U
VT
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Pictorially: Spectral form of SVD
– Best rank-k approximation in L2
Am
n
+
1u1v1 2u2v2
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SVD - Detailed outline• Motivation• Definition - properties• Interpretation
– #1: documents/terms/concepts– #2: dim. reduction– #3: picking non-zero, rectangular ‘blobs’
• Complexity• Case studies• Additional properties
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SVD - Interpretation #3• finds non-zero ‘blobs’ in a data matrix
1 1 1 0 0
2 2 2 0 0
1 1 1 0 0
5 5 5 0 0
0 0 0 2 2
0 0 0 3 30 0 0 1 1
0.18 0
0.36 0
0.18 0
0.90 0
0 0.53
0 0.800 0.27
=9.64 0
0 5.29x
0.58 0.58 0.58 0 0
0 0 0 0.71 0.71
x
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SVD - Interpretation #3• finds non-zero ‘blobs’ in a data matrix
1 1 1 0 0
2 2 2 0 0
1 1 1 0 0
5 5 5 0 0
0 0 0 2 2
0 0 0 3 30 0 0 1 1
0.18 0
0.36 0
0.18 0
0.90 0
0 0.53
0 0.800 0.27
=9.64 0
0 5.29x
0.58 0.58 0.58 0 0
0 0 0 0.71 0.71
x
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SVD - Interpretation #3• finds non-zero ‘blobs’ in a data matrix =• ‘communities’ (bi-partite cores, here)
1 1 1 0 0
2 2 2 0 0
1 1 1 0 0
5 5 5 0 0
0 0 0 2 2
0 0 0 3 30 0 0 1 1
Row 1
Row 4
Col 1
Col 3
Col 4Row 5
Row 7
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SVD - Interpretation #3• Drill: find the SVD, ‘by inspection’!• Q: rank = ??
1 1 1 0 0
1 1 1 0 0
1 1 1 0 0
0 0 0 1 1
0 0 0 1 1
= x x?? ??
??
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SVD - Interpretation #3• A: rank = 2 (2 linearly independent
rows/cols)
1 1 1 0 0
1 1 1 0 0
1 1 1 0 0
0 0 0 1 1
0 0 0 1 1
= x x??
??
?? 0
0 ??
??
??
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SVD - Interpretation #3• A: rank = 2 (2 linearly independent
rows/cols)
1 1 1 0 0
1 1 1 0 0
1 1 1 0 0
0 0 0 1 1
0 0 0 1 1
= x x?? 0
0 ??
1 0
1 0
1 0
0 1
0 11 1 1 0 0
0 0 0 1 1
orthogonal??
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SVD - Interpretation #3• column vectors: are orthogonal - but not
unit vectors:
1 1 1 0 0
1 1 1 0 0
1 1 1 0 0
0 0 0 1 1
0 0 0 1 1
= x x?? 0
0 ??
1/sqrt(3) 1/sqrt(3) 1/sqrt(3) 0 0
0 0 0 1/sqrt(2) 1/sqrt(2)
1/sqrt(3) 0
1/sqrt(3) 0
1/sqrt(3) 0
0 1/sqrt(2)
0 1/sqrt(2)
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SVD - Interpretation #3• and the singular values are:
1 1 1 0 0
1 1 1 0 0
1 1 1 0 0
0 0 0 1 1
0 0 0 1 1
= x x3 0
0 2
1/sqrt(3) 1/sqrt(3) 1/sqrt(3) 0 0
0 0 0 1/sqrt(2) 1/sqrt(2)
1/sqrt(3) 0
1/sqrt(3) 0
1/sqrt(3) 0
0 1/sqrt(2)
0 1/sqrt(2)
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SVD - Interpretation #3• Q: How to check we are correct?
1 1 1 0 0
1 1 1 0 0
1 1 1 0 0
0 0 0 1 1
0 0 0 1 1
= x x3 0
0 2
1/sqrt(3) 1/sqrt(3) 1/sqrt(3) 0 0
0 0 0 1/sqrt(2) 1/sqrt(2)
1/sqrt(3) 0
1/sqrt(3) 0
1/sqrt(3) 0
0 1/sqrt(2)
0 1/sqrt(2)
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SVD - Interpretation #3• A: SVD properties:
– matrix product should give back matrix A– matrix U should be column-orthonormal, i.e.,
columns should be unit vectors, orthogonal to each other
– ditto for matrix V– matrix should be diagonal, with positive
values
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SVD - Detailed outline• Motivation• Definition - properties• Interpretation• Complexity• Case studies• Additional properties
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SVD - Complexity• O( n * m * m) or O( n * n * m) (whichever
is less)• less work, if we just want singular values• or if we want first k singular vectors• or if the matrix is sparse [Berry]• Implemented: in any linear algebra package
(LINPACK, matlab, Splus/R, mathematica ...)
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SVD - conclusions so far• SVD: A= U VT : unique (*)• U: document-to-concept similarities• V: term-to-concept similarities• : strength of each concept• dim. reduction: keep the first few strongest
singular values (80-90% of ‘energy’)– SVD: picks up linear correlations
• SVD: picks up non-zero ‘blobs’
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References• Berry, Michael: http://www.cs.utk.edu/~lsi/• Fukunaga, K. (1990). Introduction to Statistical
Pattern Recognition, Academic Press.• Press, W. H., S. A. Teukolsky, et al. (1992).
Numerical Recipes in C, Cambridge University Press.
