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Matrix Factorization and Movie
Recommendations
Luis Serrano
@luis_likes_math
Code
https://github.com/yanneta/pytorch-tutorials/blob/master/collaborative-filtering-nn.ipynb
Yannet Interian
@interian
Question: How?
Ana
Betty
Carlos
Netflix Universe
Dana
Movie 5
Movie 2 Movie 3Movie 1
Movie 4
M1 M2 M3 M4 M5
Netflix ratings
Ana Movie 5
M1 M2 M3 M4 M5
4
M1 M2 M3 M4 M5
1 3 2 5 4
2 1 1 1 5
3 2 3 1 5
2 4 1 5 2
M
1
M
2
M
3
M
4
M
5
3 3 3 3 3
3 3 3 3 3
3 3 3 3 3
3 3 3 3 3
How do humans behave?
M
1
M
2
M
3
M
4
M
5
3 1 1 3 1
1 2 4 1 3
3 1 1 3 1
4 3 5 4 4
M
1
M
2
M
3
M
4
M
5
1 3 2 5 4
2 1 1 1 5
3 2 3 1 5
2 4 1 5 2
Dependent Rows and Columns
= =
A B C
M1 M2 M3 M4 M5
3 3 3 3 3
3 3 3 3 3
3 3 3 3 3
3 3 3 3 3
=
D
M1 = M2 = M3 = M4 = M5
M
1
M
2
M
3
M
4
M
5
3 3 3 3 3
3 3 3 3 3
3 3 3 3 3
3 3 3 3 3
How do humans behave?
M
1
M
2
M
3
M
4
M
5
3 1 1 3 1
1 2 4 1 3
3 1 1 3 1
4 3 5 4 4
M
1
M
2
M
3
M
4
M
5
1 3 2 5 4
2 1 1 1 5
3 2 3 1 5
2 4 1 5 2
Independent Rows and Columns
A B C D
M1 M2 M3 M4 M5
M1 M2 M3 M4 M5
1 3 2 5 4
2 1 1 1 5
3 2 3 1 5
2 4 1 5 2
M
1
M
2
M
3
M
4
M
5
3 3 3 3 3
3 3 3 3 3
3 3 3 3 3
3 3 3 3 3
How do humans behave?
M
1
M
2
M
3
M
4
M
5
3 1 1 3 1
1 2 4 1 3
3 1 1 3 1
4 3 5 4 4
M
1
M
2
M
3
M
4
M
5
1 3 2 5 4
2 1 1 1 5
3 2 3 1 5
2 4 1 5 2
Dependent Rows and Columns
=
A C
M1 M2 M3 M4 M5
3 1 1 3 1
1 2 4 1 3
3 1 1 3 1
4 3 5 4 4
M1 M2 M3 M4 M5
3 1 1 3 1
3 1 1 3 1
Dependent Rows and Columns
M1 M2 M3 M4 M5
3 1 1 3 1
1 2 4 1 3
3 1 1 3 1
4 3 5 4 4
M1 = M4M1 M2 M3 M4 M5
3 3
1 1
3 3
4 4
Mall Cop Observe
and
Report
Dependent Rows and Columns
M1 M2 M3 M4 M5
3 1 1 3 1
1 2 4 1 3
3 1 1 3 1
4 3 5 4 4
=
B
+
DC
M1 M2 M3 M4 M5
1 2 4 1 3
3 1 1 3 1
4 3 5 4 4
I love action
movies!
I love comedy!
I love action and
comedy movies!
Dependent Rows and Columns
M5 = Average(M2, M3)
M1 M2 M3 M4 M5
3 1 1 3 1
1 2 4 1 3
3 1 1 3 1
4 3 5 4 4
M1 M2 M3 M4 M5
1 1 1
2 4 3
1 1 1
3 5 4
Guessing ratings
Recommender Systems
= =
A B C
M1 M2 M3 M4 M5
3 3 3 3 3
3 3 3 3 3
3 3 3 ? 3
3 3 3 3 3
=
D
M1 = M2 = M3 = M4 = M5
C Movie 4
Recommender Systems
=
A C
M1 M2 M3 M4 M5
3 1 1 3 1
1 2 4 1 3
3 1 1 3
4 3 5 4 4
C Movie 5
Question: How do we figure out all these
depencencies?
Answer: Matrix Factorization
Factorization
x
Matrix Factorization
M1 M2 M3 M4 M5
3 1 1 3 1
1 2 4 1 3
3 1 1 3 1
4 3 5 4 4
this x that =
Features
ActionComedy
Sexy Canadian Ryan
Has a Sad Dog
Meryl Streep
Big Boat
Drama
Scary
Features
ActionComedy
Dot Product
ActionComedy
13
Ana
Dot Product
ActionComedy
13
Betty
Dot Product
31
ActionComedy
Dana
Categories
M1
3 1
M2
1 2
M3
1 4
M4
3 1
M5
1 3
Ana
Movie 1
3 + 1 = 3
Categories
M1
3 1
M2
1 2
M3
1 4
M4
3 1
M5
1 3
Movie 1
Betty
3 + 1 = 1
Categories
M1
3 1
M2
1 2
M3
1 4
M4
3 1
M5
1 3
Dana
Movie 5
1 + 3 = 4
Matrix Factorization
M1
3 1
M2
1 2
M3
1 4
M4
3 1
M5
1 3
=
M1 M2 M3 M4 M5
3 1 1 3 1
1 2 4 1 3
3 1 1 3 1
4 3 5 4 4
Factorization
x
Matrix
Factorization
M1 M2 M3 M4 M5
3 1 1 3 1
1 2 4 1 3
M1 M2 M3 M4 M5
3 1 1 3 1
1 2 4 1 3
3 1 1 3 1
4 3 5 4 4
Matrix
Factorization
M1 M2 M3 M4 M5
3 1 1 3 1
1 2 4 1 3
M1 M2 M3 M4 M5
3 1 1 3 1
1 2 4 1 3
3 1 1 3 1
4 3 5 4 4
Matrix
Factorization
M1 M2 M3 M4 M5
3 1 1 3 1
1 2 4 1 3
M1 M2 M3 M4 M5
3 1 1 3 1
1 2 4 1 3
3 1 1 3 1
4 3 5 4 4
Matrix
Factorization
M1 M2 M3 M4 M5
3 1 1 3 1
1 2 4 1 3
M1 M2 M3 M4 M5
3 1 1 3 1
1 2 4 1 3
3 1 1 3 1
4 3 5 4 4
Matrix
Factorization
M1 M2 M3 M4 M5
3 1 1 3 1
1 2 4 1 3
M1 M2 M3 M4 M5
3 1 1 3 1
1 2 4 1 3
3 1 1 3 1
4 3 5 4 4
Storage
Matrix
Factorization
M1 M2 M3 M4 M5
3 1 1 3 1
1 2 4 1 3
M1 M2 M3 M4 M5
3 1 1 3 1
1 2 4 1 3
3 1 1 3 1
4 3 5 4 4
Matrix
Factorization
2000 x 1000
2M entries
100 x 1000
100K entries
2000 x 100
200K entries
2000 Users
1000
Movies
100 Features
100
Features
2000 Users
1000
Movies
Quiz: How many parameters (arrows)?
20 parameters 18 parameters
2000 users
1000 movies
2000 x 1000 = 2,000,000 parameters
Quiz: How many parameters?
2000 users
1000 movies
300,000 parameters
2000x100 = 200,000 parameters
100 x 1000 = 100,000 parameters
Quiz: How many parameters?
100 features
Storage?
2M parameters 300K parameters
How to find the right factorization?
?
Nope
Training models
?
Nope
Training models
Training models
?
Close
enough!
Training models
Gradient Descent
Matrix
Factorization
M1 M2 M3 M4 M5
3 1 1 3 1
1 2 4 1 3
1 0
0 1
1 0
1 1
M1 M2 M3 M4 M5
3 1 1 3 1
1 2 4 1 3
3 1 1 3 1
4 3 5 4 4
1
0
M1 M2 M3 M4 M5
F1 1.2 3.1 0.3 2.5 0.2
F2 2.4 1.5 4.4 0.4 1.1
M1 M2 M3 M4 M5 M1 M2 M3 M4 M5
3 1 1 3 1
1 2 4 1 3
3 1 1 3 1
4 3 5 4 4
F1 F2
0.2 0.5
0.3 0.4
0.7 0.8
0.4 0.5
M1 M2 M3 M4 M5
1.44 1.37 2.26 0.7 0.59
1.32 1.53 1.85 0.91 0.5
2.76 3.37 3.73 2.07 1.02
1.68 1.99 2.32 1.2 0.63
1.2 x 0.2 + 2.4 x 0.5 = 1.44
M1 M2 M3 M4 M5
F1 1.2 3.1 0.3 2.5 0.2
F2 2.4 1.5 4.4 0.4 1.1
M1 M2 M3 M4 M5
1.44 1.37 2.26 0.7 0.59
1.32 1.53 1.85 0.91 0.5
2.76 3.37 3.73 2.07 1.02
1.68 1.99 2.32 1.2 0.63
M1 M2 M3 M4 M5
3 1 1 3 1
1 2 4 1 3
3 1 1 3 1
4 3 5 4 4
F1 F2
0.2 0.5
0.3 0.4
0.7 0.8
0.4 0.5
F1 F2
0.2 0.5
0.3 0.4
0.7 0.8
0.4 0.5
M1 M2 M3 M4 M5
F1 1.2 3.1 0.3 2.5 0.2
F2 2.4 1.5 4.4 0.4 1.1
M1 M2 M3 M4 M5
1.44
M1 M2 M3 M4 M5
3 1 1 3 1
1 2 4 1 3
3 1 1 3 1
4 3 5 4 4
1.2 x 0.2 + 2.4 x 0.5 = 1.44
F1 F2
0.3 0.6
0.3 0.4
0.7 0.8
0.4 0.5
M1 M2 M3 M4 M5
F1 1.4 3.1 0.3 2.5 0.2
F2 2.5 1.5 4.4 0.4 1.1
M1 M2 M3 M4 M5 M1 M2 M3 M4 M5
3 1 1 3 1
1 2 4 1 3
3 1 1 3 1
4 3 5 4 4
F1 F2
0.3 0.6
0.3 0.4
0.7 0.8
0.4 0.5
M1 M2 M3 M4 M5
F1 1.4 3.1 0.3 2.5 0.2
F2 2.5 1.5 4.4 0.4 1.1
M1 M2 M3 M4 M5
1.92
M1 M2 M3 M4 M5
3 1 1 3 1
1 2 4 1 3
3 1 1 3 1
4 3 5 4 4
1.4 x 0.3 + 2.5 x 0.6 = 1.92
F1 F2
0.3 0.6
0.3 0.4
0.7 0.8
0.4 0.5
M1 M2 M3 M4 M5
F1 1.4 3.1 0.3 2.5 0.2
F2 2.5 1.5 4.4 0.4 1.1
M1 M2 M3 M4 M5
1.92 1.83
M1 M2 M3 M4 M5
3 1 1 3 1
1 2 4 1 3
3 1 1 3 1
4 3 5 4 4
1.4 x 0.3 + 2.5 x 0.6 = 1.92
M1 M2 M3 M4 M5
F1 1.4 3.1 0.3 2.5 0.2
F2 2.5 1.5 4.4 0.4 1.1
F1 F2
0.3 0.6
0.3 0.4
0.7 0.8
0.4 0.5
M1 M2 M3 M4 M5
1.92 1.83
M1 M2 M3 M4 M5
3 1 1 3 1
1 2 4 1 3
3 1 1 3 1
4 3 5 4 4
3.1 x 0.3 + 1.5 x 0.6 = 1.83
Error function
?
Nope!
Training models
You were
wrong by
10.3
?
Training models
You were
wrong by
5.32
?
Training models
Training models
You were
wrong by
1.23
?
Matrix
Factorization
M1 M2 M3 M4 M5
3 1 1 3 1
1 2 4 1 3
1 0
0 1
1 0
1 1
M1 M2 M3 M4 M5
3 1 1 3 1
1 2 4 1 3
3 1 1 3 1
4 3 5 4 4
F1 F2
0.2 0.5
0.3 0.4
0.7 0.8
0.4 0.5
M1 M2 M3 M4 M5
F1 1.2 3.1 0.3 2.5 0.2
F2 2.4 1.5 4.4 0.4 1.1
M1 M2 M3 M4 M5
1.44
M1 M2 M3 M4 M5
3 1 1 3 1
1 2 4 1 3
3 1 1 3 1
4 3 5 4 4
Error Function
Error = (3 - 1.44)
2
F1 F2
0.2 0.5
0.3 0.4
0.7 0.8
0.4 0.5
M1 M2 M3 M4 M5
F1 1.2 3.1 0.3 2.5 0.2
F2 2.4 1.5 4.4 0.4 1.1
M1 M2 M3 M4 M5
1.44 1.37
M1 M2 M3 M4 M5
3 1 1 3 1
1 2 4 1 3
3 1 1 3 1
4 3 5 4 4
Error Function
Error = (3 - 1.44)
2
+ (1 - 1.37)
2
+ …
Derivative
Summary: How to use this?
M1 M2 M3 M4 M5
F1 3 1 1 3 1
F2 1 2 4 1 3
F1 F2
1 0
0 1
1 0
1 1
M1 M2 M3 M4 M5
3 1 1
1 4 1
3 1 3 1
3 4 4
M1 M2 M3 M4 M5
1 3
2 3
1
4 5
D
M3
Thank you!
Luis Serrano
@luis_likes_math
www.youtube.com/c/LuisSerrano
luisguiserrano.github.io
www.udacity.com
Matrix factorization

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