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Singular Value Decomposition
Luis Serrano
Announcements
github.com/luisguiserrano/singular_value_decomposition
https://www.manning.com/books/grokking-machine-learning
Discount code: serranoyt
Grokking Machine
Learning
By Luis G. Serrano
Transformations
Transformations
Stretch (or compress) horizontally
Transformations
Stretch (or compress) horizontally
Transformations
Stretch (or compress) horizontally
Transformations
Stretch (or compress) horizontally
Transformations
Stretch (or compress) vertically
Transformations
Stretch (or compress) vertically
Transformations
Stretch (or compress) vertically
Transformations
Stretch (or compress) vertically
Transformations
Rotate
Puzzle (easy)
Puzzle (easy)
Puzzle (easy)
Puzzle (easy)
Puzzle (easy)
Puzzle (hard)
Puzzle (hard)
Puzzle (hard)
Puzzle (hard)
Puzzle (hard)
Solution
Solution
Solution
Solution
Solution
Solution
Solution
Linear transformations
-7
-5
-3
-1
1
3
5
7
-7-6-5-4-3-2-1 0 1 2 3 4 5 6 7
-7
-5
-3
-1
1
3
5
7
-7-6-5-4-3-2-1 0 1 2 3 4 5 6 7
What does this have to do with matrices?
(p,q) (3p+0q, 4p+5q)
(1,0) (3, 4)
(0,1) (0, 5)
(-1,0)
(0,-1)
(-3, -4)
(0, -5)
3 0

4 5[ ]A =
-7
-5
-3
-1
1
3
5
7
-7-6-5-4-3-2-1 0 1 2 3 4 5 6 7
Rotation matrices

cos(θ) −sin(θ)
sin(θ) cos(θ)[ ]
θ
-7
-5
-3
-1
1
3
5
7
-7-6-5-4-3-2-1 0 1 2 3 4 5 6 7
Stretching matrices

σ1 0
0 σ2
[ ]
σ1
σ2
Stretching matrices

σ1 0
0 σ2
[ ]
-7
-5
-3
-1
1
3
5
7
-7-6-5-4-3-2-1 0 1 2 3 4 5 6 7
σ1
σ2
Stretching matrices

σ1 0
0 σ2
[ ]
-7
-5
-3
-1
1
3
5
7
-7-6-5-4-3-2-1 0 1 2 3 4 5 6 7
σ1
σ2
-7
-5
-3
-1
1
3
5
7
-7-6-5-4-3-2-1 0 1 2 3 4 5 6 7
-7
-5
-3
-1
1
3
5
7
-7-6-5-4-3-2-1 0 1 2 3 4 5 6 7
What does this have to do with matrices?
3 0

4 5[ ]A =

cos(θ) sin(θ)
−sin(θ) cos(θ)[ ]

cos(ϕ) sin(ϕ)
−sin(ϕ) cos(ϕ)[ ]

σ1 0
0 σ2
[ ]
Singular value decomposition
3 0

4 5[ ] 
cos(θ) sin(θ)
−sin(θ) cos(θ)[ ] 
cos(ϕ) sin(ϕ)
−sin(ϕ) cos(ϕ)[ ]
σ1 0
0 σ2
[ ]=
A = UΣV†
-7
-5
-3
-1
1
3
5
7
-7-6-5-4-3-2-1 0 1 2 3 4 5 6 7
SVD A = UΣV†

3 5 0
0 5[ ]
1/ 10 −3/ 10
3/ 10 1/ 10[ ]

0.7071 0.7071
−0.7071 0.7071[ ]
6.708 0
0 2.236[ ]
0.316 −0.949
0.949 0.316[ ]

1/ 2 1/ 2
−1/ 2 1/ 2[ ]
Rotation of θ = −
π
4
= − 45o
3 0

4 5[ ] =
-7
-5
-3
-1
1
3
5
7
-7-6-5-4-3-2-1 0 1 2 3 4 5 6 7
Rotation

1/ 2 1/ 2
−1/ 2 1/ 2[ ]
3 5 0
0 5[ ]
1/ 10 −3/ 10
3/ 10 1/ 10[ ]

0.7071 0.7071
−0.7071 0.7071[ ]
6.708 0
0 2.236[ ]
0.316 −0.949
0.949 0.316[ ]
A = UΣV†
Rotation of θ = −
π
4
= − 45o
3 0

4 5[ ] =
-7
-5
-3
-1
1
3
5
7
-7-6-5-4-3-2-1 0 1 2 3 4 5 6 7
Rotation

1/ 2 1/ 2
−1/ 2 1/ 2[ ]
3 5 0
0 5[ ]
1/ 10 −3/ 10
3/ 10 1/ 10[ ]

0.7071 0.7071
−0.7071 0.7071[ ]
6.708 0
0 2.236[ ]
0.316 −0.949
0.949 0.316[ ]
A = UΣV†
Horizontal scaling by 3 5
3 0

4 5[ ] =
-7
-5
-3
-1
1
3
5
7
-7-6-5-4-3-2-1 0 1 2 3 4 5 6 7
Scaling

1/ 2 1/ 2
−1/ 2 1/ 2[ ]
3 5 0
0 5[ ]
1/ 10 −3/ 10
3/ 10 1/ 10[ ]

0.7071 0.7071
−0.7071 0.7071[ ]
6.708 0
0 2.236[ ]
0.316 −0.949
0.949 0.316[ ]
A = UΣV†
Horizontal scaling by 3 5
Vertical scaling by 5
3 0

4 5[ ] =
-7
-5
-3
-1
1
3
5
7
-7-6-5-4-3-2-1 0 1 2 3 4 5 6 7
Rotation

1/ 2 1/ 2
−1/ 2 1/ 2[ ]
3 5 0
0 5[ ]
1/ 10 −3/ 10
3/ 10 1/ 10[ ]

0.7071 0.7071
−0.7071 0.7071[ ]
6.708 0
0 2.236[ ]
0.316 −0.949
0.949 0.316[ ]
A = UΣV†
Vertical scaling by 5
3 0

4 5[ ] =
-7
-5
-3
-1
1
3
5
7
-7-6-5-4-3-2-1 0 1 2 3 4 5 6 7
Rotation

1/ 2 1/ 2
−1/ 2 1/ 2[ ]
3 5 0
0 5[ ]
1/ 10 −3/ 10
3/ 10 1/ 10[ ]

0.7071 0.7071
−0.7071 0.7071[ ]
6.708 0
0 2.236[ ]
0.316 −0.949
0.949 0.316[ ]
A = UΣV†
Rotation of θ = arctan(3) = 71.72o
3 0

4 5[ ] =
-7
-5
-3
-1
1
3
5
7
-7-6-5-4-3-2-1 0 1 2 3 4 5 6 7
Rotation

1/ 2 1/ 2
−1/ 2 1/ 2[ ]
3 5 0
0 5[ ]
1/ 10 −3/ 10
3/ 10 1/ 10[ ]

0.7071 0.7071
−0.7071 0.7071[ ]
6.708 0
0 2.236[ ]
0.316 −0.949
0.949 0.316[ ]
A = UΣV†
Rotation of θ = arctan(3) = 71.72o
3 0

4 5[ ] =
-7
-5
-3
-1
1
3
5
7
-7 -5 -3 -1 1 3 5 7
-7
-5
-3
-1
1
3
5
7
-7
-5
-3
-1
1
3
5
7
-7
-5
-3
-1
1
3
5
7
-7 -5 -3 -1 1 3 5 7
A = UΣV†
A
Σ
V† U
Dimensionality reduction
1 2 3 4
-1 -2 -3 -4
2 4 6 8
10 20 30 40
1 2 3 4
1
-1
2
10
Difference between these two matrices?
1 2 3 4
-1 -2 -3 -4
2 4 6 8
10 20 30 40
1 2 3 4
1
-1
2
10
Difference between these two matrices?
3 1 4 1
5 9 2 6
5 3 5 8
9 7 9 3
? ? ? ?
?
?
?
?
=
1 2 3 4
-1 -2 -3 -4
2 4 6 8
10 20 30 40
1 2 3 4
1
-1
2
10
Rank 1 matrices
16 numbers 8 numbers
=
3 1 4 1
5 9 2 6
5 3 5 8
9 7 9 3
? ? ? ?
?
?
?
?
Higher rank matrices
16 numbers
Rank of a matrix
=
Rank 1
=
Rank 2
=
Rank 3
=
Rank 4
∼
3 1 4 1
5 9 2 6
5 3 5 8
9 7 9 3
? ? ? ?
?
?
?
?
Approximation by a rank one matrix
0
1
2
3
4
0 1 2 3 4
U1 U2 U3 U4
0
1
2
3
4
0 1 2 3 4
V1
V2
V3
V4
0
1
2
3
4
0 1 2 3 4
σ1
σ2
σ3
σ4σ4
= U1 U2 U3 U4
V1
V2
V3
V4
σ1
σ2
σ3
σ4
0
1
2
3
4
0 1 2 3 4
A
0
1
2
3
4
0 1 2 3 4
U1 U2 U3 U4
0
1
2
3
4
0 1 2 3 4
V1
V2
V3
V4
0
1
2
3
4
0 1 2 3 4
σ1
σ2
σ3
σ4
U1
U2 U3 U4
V1
V2
V3
V4
σ1
σ2
σ3
σ4
+
=
0
1
2
3
4
0 1 2 3 4
A
0
1
2
3
4
0 1 2 3 4
U1 U2 U3 U4
0
1
2
3
4
0 1 2 3 4
V1
V2
V3
V4
0
1
2
3
4
0 1 2 3 4
σ1
σ2
σ3
σ4
U1 U2
U3 U4
V1 V2
V3
V4
σ1 σ2
σ3
σ4
+
=
0
1
2
3
4
0 1 2 3 4
A
0
1
2
3
4
0 1 2 3 4
U1 U2 U3 U4
0
1
2
3
4
0 1 2 3 4
V1
V2
V3
V4
0
1
2
3
4
0 1 2 3 4
σ1
σ2
σ3
σ4
U1 U2 U3
U4
V1 V2 V3
V4
σ1 σ2 σ3
σ4
+
=
+
0
1
2
3
4
0 1 2 3 4
A
0
1
2
3
4
0 1 2 3 4
U1 U2 U3 U4
0
1
2
3
4
0 1 2 3 4
V1
V2
V3
V4
0
1
2
3
4
0 1 2 3 4
σ1
σ2
σ3
σ4
U1 U2 U3 U4V1 V2 V3 V4σ1 σ2 σ3 σ4+
=
+ +
Rank 1 Rank 1 Rank 1 Rank 1
TinySmallMediumLarge
0
1
2
3
4
0 1 2 3 4
A
0
1
2
3
4
0 1 2 3 4
U1 U2 U3 U4
0
1
2
3
4
0 1 2 3 4
V1
V2
V3
V4
0
1
2
3
4
0 1 2 3 4
σ1
σ2
σ3
σ4
U1 U2V1 V2σ1 σ2+
=
Rank 1 Rank 1
MediumLarge
0
1
2
3
4
0 1 2 3 4
A
3 1 4 1
5 9 2 6
5 3 5 8
9 7 9 3
0.15+
Tiny
0.79 -0.29 -0.54 -0.04
-0.89
-0.23
0.15
0.36
0
1
2
3
4
0 1 2 3 4
21.2
6.4
4.9
0.15
=
-0.55 -0.52 -0.49 -0.43
0.26 -0.4 0.65 -0.59
0.07 0.7 -0.22 -0.68
0.79 -0.29 -0.54 -0.04
-0.21 0.37 -0.13 -0.89
-0.52 -0.7 0.43 -0.23
-0.48 -0.21 -0.84 0.15
-0.67 0.57 0.31 0.36
4.9+ 0.07 0.7 -0.22 -0.68
-0.13
0.43
-0.84
0.31
Small
6.4+ 0.26 -0.4 0.65 -0.59
0.37
-0.7
-0.21
0.57
Medium
21.2 -0.55 -0.52 -0.49 -0.43
-0.21
-0.52
-0.48
-0.67
Large
2.51 2.37 2.22 1.97
6.07 5,72 5.37 4.77
5.63 5,31 4.99 4.43
7.88 7.43 6.98 6.19
3.15 1.41 3.79 0.56
4.87 7.53 2.44 7.43
5.28 5.85 4.12 5.22
8.85 5.96 9.36 4.03
3.1 0.96 3.93 0.99
5.03 8.99 1.98 6
4.98 3.01 5.01 8
8.96 7.02 9.03 3
3 1 4 1
5 9 2 6
5 3 5 8
9 7 9 3
Rank of a matrix
=
Rank 1
=
Rank 2
=
Rank 3
=
Rank 4
Dimensionality reduction
-7
-5
-3
-1
1
3
5
7
-7 -5 -3 -1 1 3 5 7

1.8 1.2
4.4 4.6[ ]
A
-7
-5
-3
-1
1
3
5
7
-7 -5 -3 -1 1 3 5 7
-7
-5
-3
-1
1
3
5
7
-7
-5
-3
-1
1
3
5
7
A = UΣV† 
0.316 −0.949
0.949 0.316[ ]U

6.71 0
0 0.44[ ]Σ

0.7071 0.7071
−0.7071 0.7071[ ] V†
6.71
0.44
-7
-5
-3
-1
1
3
5
7
-7 -5 -3 -1 1 3 5 7
-7
-5
-3
-1
1
3
5
7
-7
-5
-3
-1
1
3
5
7

6.71 0
0 0.44[ ]

1.8 1.2
4.4 4.6[ ]
A = UΣV†
A
Σ
-7
-5
-3
-1
1
3
5
7
-7 -5 -3 -1 1 3 5 7
6.71
0.44
0
0

0.7071 0.7071
−0.7071 0.7071[ ] V† 
0.316 −0.949
0.949 0.316[ ]U
-7
-5
-3
-1
1
3
5
7
-7 -5 -3 -1 1 3 5 7
-7
-5
-3
-1
1
3
5
7
-7
-5
-3
-1
1
3
5
7

1.8 1.2
4.4 4.6[ ]
A = UΣV†
A
Σ
-7
-5
-3
-1
1
3
5
7
-7 -5 -3 -1 1 3 5 7
6.71
[ 
6.71 0
0 0.44 ]
0.44
0
0

0.316 −0.949
0.949 0.316[ ]U
0.7071 0.7071
−0.7071 0.7071[ ] V†
-7
-5
-3
-1
1
3
5
7
-7 -5 -3 -1 1 3 5 7
-7
-5
-3
-1
1
3
5
7
-7
-5
-3
-1
1
3
5
7

1.8 1.2
4.4 4.6[ ]
A = UΣV†
A
Σ
-7
-5
-3
-1
1
3
5
7
-7 -5 -3 -1 1 3 5 7
6.71
[ 
6.71 0
0 0.44 ]
0.44
0
0

1.5 1.5
4.5 4.5[ ]

0.316 −0.949
0.949 0.316[ ]U
0.7071 0.7071
−0.7071 0.7071[ ] V†
-7
-5
-3
-1
1
3
5
7
-7 -5 -3 -1 1 3 5 7
-7
-5
-3
-1
1
3
5
7
-7 -5 -3 -1 1 3 5 7

1.5 1.5
4.5 4.5[ ]
Rank 1
-7
-5
-3
-1
1
3
5
7
-7 -5 -3 -1 1 3 5 7

1.8 1.2
4.4 4.6[ ]
Rank 2
-7
-5
-3
-1
1
3
5
7
-7 -5 -3 -1 1 3 5 7

1.5 1.5
4.5 4.5[ ]
Rank 1
1 2 3 4
-1 -2 -3 -4
2 4 6 8
10 20 30 40
1 2 3 4
1
-1
2
10
Rank 1 matrices
=
1 2 3 4
-1 -2 -3 -4
2 4 6 8
10 20 30 40
1 2 3 4
1
-1
2
10
Rank 1 matrices
16 numbers 8 numbers
=
3 1 4 1
5 9 2 6
5 3 5 8
9 7 9 3
? ? ? ?
?
?
?
?
Higher rank matrices
16 numbers
Approximation by rank one matrices
= + + …
3 1 4 1
5 9 2 6
5 3 5 8
9 7 9 3
U
0
1
2
3
4
0 1 2 3 4
V†
Σ= U1 U2 U3 U4
V1
V2
V3
V4
0
1
2
3
4
0 1 2 3 4
σ1
σ2
σ3
σ4
0
1
2
3
4
0 1 2 3 4
A
0
1
2
3
4
0 1 2 3 4
U1 U2 U3 U4
0
1
2
3
4
0 1 2 3 4
V1
V2
V3
V4
0
1
2
3
4
0 1 2 3 4
σ1
σ2
σ3
σ4σ4
= U1 U2 U3 U4
V1
V2
V3
V4
σ1
σ2
σ3
σ4
0
1
2
3
4
0 1 2 3 4
A
0
1
2
3
4
0 1 2 3 4
U1 U2 U3 U4
0
1
2
3
4
0 1 2 3 4
V1
V2
V3
V4
0
1
2
3
4
0 1 2 3 4
σ1
σ2
σ3
σ4
U1
U2 U3 U4
V1
V2
V3
V4
σ1
σ2
σ3
σ4
+
=
0
1
2
3
4
0 1 2 3 4
A
0
1
2
3
4
0 1 2 3 4
U1 U2 U3 U4
0
1
2
3
4
0 1 2 3 4
V1
V2
V3
V4
0
1
2
3
4
0 1 2 3 4
σ1
σ2
σ3
σ4
U1 U2
U3 U4
V1 V2
V3
V4
σ1 σ2
σ3
σ4
+
=
0
1
2
3
4
0 1 2 3 4
A
0
1
2
3
4
0 1 2 3 4
U1 U2 U3 U4
0
1
2
3
4
0 1 2 3 4
V1
V2
V3
V4
0
1
2
3
4
0 1 2 3 4
σ1
σ2
σ3
σ4
U1 U2 U3
U4
V1 V2 V3
V4
σ1 σ2 σ3
σ4
+
=
+
0
1
2
3
4
0 1 2 3 4
A
0
1
2
3
4
0 1 2 3 4
U1 U2 U3 U4
0
1
2
3
4
0 1 2 3 4
V1
V2
V3
V4
0
1
2
3
4
0 1 2 3 4
σ1
σ2
σ3
σ4
U1 U2 U3 U4V1 V2 V3 V4σ1 σ2 σ3 σ4+
=
+ +
Rank 1 Rank 1 Rank 1 Rank 1
TinySmallMediumLarge
0
1
2
3
4
0 1 2 3 4
A
0
1
2
3
4
0 1 2 3 4
U1 U2 U3 U4
0
1
2
3
4
0 1 2 3 4
V1
V2
V3
V4
0
1
2
3
4
0 1 2 3 4
σ1
σ2
σ3
σ4
U1 U2V1 V2σ1 σ2+
=
Rank 1 Rank 1
MediumLarge
0
1
2
3
4
0 1 2 3 4
A
Rectangular matrices
0
1
2
3
4
0 1 2 3 4
U1 U2 U3 U4=
0
1
2
3
4
0 1 2 3 4 5 6
A
V1
V2
V3
V4
V5
V6
σ1
σ2
σ3
σ4
0 0
0 0
0 0
0 0
No square matrix? No problem!
Image compression
0 1 1 0 1 1 0
1 1 1 1 1 1 1
1 1 1 1 1 1 1
0 1 1 1 1 1 0
0 0 1 1 1 0 0
0 0 0 1 0 0 0
Rank 4
www.github.com/luisguiserrano/singular_value_decomposition
Thank you!
Similar videos on dimensionality reduction
Matrix Factorization Principal Component Analysis
https://www.manning.com/books/grokking-machine-learning
Discount code: serranoyt
Grokking Machine
Learning
By Luis G. Serrano
Thank you!
@luis_likes_math
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