1. Jia-Bin Huang Virginia Tech
Recommender Systems
Jia-Bin Huang
Virginia Tech
ECE-5424G / CS-5824
Spring 2019

2. Administrative
HW 4 due April 10

3. Unsupervised Learning
Clustering, K-Mean
Expectation maximization
Dimensionality reduction
Anomaly detection
Recommendation system

4. Motivating example: Monitoring machines in a data center
𝑥 2 (Memory use)
𝑥 1 (CPU load)
𝑥 2 (Memory use)
𝑥 1 (CPU load)

5. Multivariate Gaussian (normal) distribution
𝑥∈ 𝑅 𝑛 . Don’t model 𝑝 𝑥 1 ,𝑝 𝑥 2 , ⋯ separately
Model 𝑝 𝑥 all in one go.
Parameters: 𝜇∈ 𝑅 𝑛 , Σ∈ 𝑅 𝑛×𝑛 (covariance matrix)
𝑝 𝑥;𝜇, Σ = 1 2𝜋 𝑛/2 Σ 1/2 exp − 𝑥−𝜇 ⊤ Σ −1 (𝑥−𝜇)

6. Multivariate Gaussian (normal) examples
Σ =
Σ =
Σ =
𝑥 2
𝑥 2
𝑥 2
𝑥 1
𝑥 1
𝑥 1

7. Multivariate Gaussian (normal) examples
Σ =
Σ =
Σ =
𝑥 2
𝑥 2
𝑥 2
𝑥 1
𝑥 1
𝑥 1

8. Multivariate Gaussian (normal) examples
Σ =
Σ =
Σ =
𝑥 2
𝑥 2
𝑥 2
𝑥 1
𝑥 1
𝑥 1

9. Anomaly detection using the multivariate Gaussian distribution
Fit model 𝑝 𝑥 by setting
𝜇= 1 𝑚 𝑖=1 𝑚 𝑥 (𝑖)
Σ= 1 𝑚 𝑖=1 𝑚 (𝑥 (𝑖) −𝜇)(𝑥 (𝑖) − 𝜇) ⊤
2 Give a new example 𝑥, compute
𝑝 𝑥;𝜇, Σ = 1 2𝜋 𝑛/2 Σ 1/2 exp − 𝑥−𝜇 ⊤ Σ −1 (𝑥−𝜇)
Flag an anomaly if 𝑝 𝑥 <𝜖

10. Automatically captures correlations between features
Original model 𝑝 𝑥 1 ; 𝜇 1 , 𝜎 1 2 𝑝 𝑥 2 ; 𝜇 2 , 𝜎 2 2 ⋯𝑝 𝑥 𝑛 ; 𝜇 𝑛 , 𝜎 𝑛 2 Manually create features to capture anomalies where 𝑥 1 , 𝑥 2 take unusual combinations of values Computationally cheaper (alternatively, scales better) OK even if training set size is small
Original model
𝑝 𝑥;𝜇, Σ = 1 2𝜋 𝑛/2 Σ 1/2 exp − 𝑥−𝜇 ⊤ Σ −1 (𝑥−𝜇) exp − 𝑥−𝜇 ⊤ Σ −1 (𝑥−𝜇)
Automatically captures correlations between features
Computationally more expensive
Must have 𝑚>𝑛 or else Σ is non- invertible

11. Recommender Systems Motivation Problem formulation
Content-based recommendations
Collaborative filtering
Mean normalization

12. Recommender Systems Motivation Problem formulation
Content-based recommendations
Collaborative filtering
Mean normalization

13. You may also like..?

15. Recommender Systems Motivation Problem formulation
Content-based recommendations
Collaborative filtering
Mean normalization

16. Example: Predicting movie ratings
User rates movies using zero to five stars
𝑛 𝑢 = no. users
𝑛 𝑚 = no. movies
𝑟 𝑖,𝑗 =1 if user 𝑗 has rated movie 𝑖
𝑦 (𝑖,𝑗) = rating given by user 𝑗 to movie 𝑖
Movie
Alice (1)
Bob (2)
Carol (3)
Dave (4)
Love at last 5
Romance forever ?
Cute puppies of love 4
Nonstop car chases
Swords vs. karate

17. Recommender Systems Motivation Problem formulation
Content-based recommendations
Collaborative filtering
Mean normalization

18. Content-based recommender systems
Movie
Alice (1)
Bob (2)
Carol (3)
Dave (4)
𝑥 1
(romance)
𝑥 2
(action)
Love at last 5
0.9
Romance forever ?
1.0
0.01
Cute puppies of love 4
0.99
Nonstop car chases
0.1
Swords vs. karate
For each user 𝑗, learn a parameter 𝜃 (𝑗) ∈ 𝑅 3 .
Predict user 𝑗 as rating movie 𝑖 with (𝜃 𝑗 ) ⊤ 𝑥 (𝑖) stars.

19. Content-based recommender systems
Movie
Alice (1)
Bob (2)
Carol (3)
Dave (4)
𝑥 1
(romance)
𝑥 2
(action)
Love at last 5
0.9
Romance forever ?
1.0
0.01
Cute puppies of love 4
0.99
Nonstop car chases
0.1
Swords vs. karate
𝑥 (3) =
𝜃 1 =
(𝜃 1 ) ⊤ 𝑥 (3) =5∗0.99=4.95
For each user 𝑗, learn a parameter 𝜃 (𝑗) ∈ 𝑅 3 .
Predict user 𝑗 as rating movie 𝑖 with (𝜃 𝑗 ) ⊤ 𝑥 (𝑖) stars.

20. Problem formulation
𝑟 𝑖,𝑗 =1 if user 𝑗 has rated movie 𝑖
𝑦 (𝑖,𝑗) = rating given by user 𝑗 to movie 𝑖
𝜃 (𝑗) = parameter vector for user 𝑗
𝑥 (𝑖) = feature vector for user 𝑖
For each user 𝑗, predicted rating: (𝜃 𝑗 ) ⊤ 𝑥 (𝑖)
𝑚 (𝑗) = no. of movies rated by user j
Goal: learn 𝜃 (𝑗) :
min 𝜃 (𝑗) 𝑚 (𝑗) 𝑖:𝑟 𝑖,𝑗 = (𝜃 𝑗 ) ⊤ 𝑥 𝑖 − 𝑦 𝑖,𝑗 𝜆 2 𝑚 (𝑗) 𝑘=1 𝑛 𝜃 𝑘 𝑗 2

21. Optimization objective
Learn 𝜃 𝑗 (parameter for user 𝑗):
min 𝜃 (𝑗) 𝑖:𝑟 𝑖,𝑗 = (𝜃 𝑗 ) ⊤ 𝑥 𝑖 − 𝑦 𝑖,𝑗 𝜆 2 𝑘=1 𝑛 𝜃 𝑘 𝑗 2
Learn 𝜃 1 , 𝜃 2 , ⋯, 𝜃 𝑛 𝑢 :
min 𝜃 1 , 𝜃 2 , ⋯, 𝜃 𝑛 𝑢 𝑗=1 𝑛 𝑢 𝑖:𝑟 𝑖,𝑗 = (𝜃 𝑗 ) ⊤ 𝑥 𝑖 − 𝑦 𝑖,𝑗 𝜆 2 𝑗=1 𝑛 𝑢 𝑘=1 𝑛 𝜃 𝑘 𝑗 2

22. Optimization algorithm
min 𝜃 (𝑗) 𝑗=1 𝑛 𝑢 𝑖:𝑟 𝑖,𝑗 = (𝜃 𝑗 ) ⊤ 𝑥 𝑖 − 𝑦 𝑖,𝑗 𝜆 2 𝑗=1 𝑛 𝑢 𝑘=1 𝑛 𝜃 𝑘 𝑗 2
Gradient descent update:
𝜃 𝑘 𝑗 ≔ 𝜃 𝑘 𝑗 −𝛼 𝑖:𝑟 𝑖,𝑗 = 𝜃 𝑗 ⊤ 𝑥 𝑖 − 𝑦 𝑖,𝑗 𝑥 𝑘 𝑖 (for 𝑘=0)
𝜃 𝑘 𝑗 ≔ 𝜃 𝑘 𝑗 −𝛼 𝑖:𝑟 𝑖,𝑗 =1 ( 𝜃 𝑗 ⊤ 𝑥 𝑖 − 𝑦 𝑖,𝑗 ) 𝑥 𝑘 𝑖 +𝜆 𝜃 𝑘 (𝑗) (for 𝑘≠0)

23. Recommender Systems Motivation Problem formulation
Content-based recommendations
Collaborative filtering
Mean normalization

24. Problem motivation 5 0.9 ? 1.0 0.01 4 0.99 0.1 Movie Alice (1) Bob (2)
Carol (3)
Dave (4)
𝑥 1
(romance)
𝑥 2
(action)
Love at last 5
0.9
Romance forever ?
1.0
0.01
Cute puppies of love 4
0.99
Nonstop car chases
0.1
Swords vs. karate

25. Problem motivation 𝜃 1 = 0 5 0 𝜃 2 = 0 5 0 𝜃 3 = 0 0 5 𝜃 4 = 0 0 5
Movie
Alice (1)
Bob (2)
Carol (3)
Dave (4)
𝑥 1
(romance)
𝑥 2
(action)
Love at last 5 ?
Romance forever
Cute puppies of love 4
Nonstop car chases
Swords vs. karate
𝜃 1 =
𝜃 2 =
𝜃 3 =
𝜃 4 =
𝑥 1 = ? ? ?

26. Optimization algorithm
Given 𝜃 1 , 𝜃 2 , ⋯, 𝜃 𝑛 𝑢 , to learn 𝑥 (𝑖) :
min 𝑥 (𝑖) 𝑗:𝑟 𝑖,𝑗 = (𝜃 𝑗 ) ⊤ 𝑥 𝑖 − 𝑦 𝑖,𝑗 𝜆 2 𝑘=1 𝑛 𝑥 𝑘 (𝑖) 2
Given 𝜃 1 , 𝜃 2 , ⋯, 𝜃 𝑛 𝑢 , to learn 𝑥 (1) , 𝑥 (2) , ⋯, 𝑥 ( 𝑛 𝑚 ) :
min 𝑥 (1) , 𝑥 (2) , ⋯, 𝑥 ( 𝑛 𝑚 ) 𝑖=1 𝑛 𝑚 𝑗:𝑟 𝑖,𝑗 = (𝜃 𝑗 ) ⊤ 𝑥 𝑖 − 𝑦 𝑖,𝑗 𝜆 2 𝑖=1 𝑛 𝑚 𝑘=1 𝑛 𝑥 𝑘 (𝑖) 2

27. Collaborative filtering
Given 𝑥 1 , 𝑥 2 , ⋯, 𝑥 𝑛 𝑚 (and movie ratings),
Can estimate 𝜃 1 , 𝜃 2 , ⋯, 𝜃 𝑛 𝑢
Given 𝜃 1 , 𝜃 2 , ⋯, 𝜃 𝑛 𝑢
Can estimate 𝑥 1 , 𝑥 2 , ⋯, 𝑥 𝑛 𝑚

28. Collaborative filtering optimization objective
Given 𝑥 1 , 𝑥 2 , ⋯, 𝑥 𝑛 𝑚 , estimate 𝜃 1 , 𝜃 2 , ⋯, 𝜃 𝑛 𝑢
min 𝜃 1 , 𝜃 2 , ⋯, 𝜃 𝑛 𝑢 𝑗=1 𝑛 𝑢 𝑖:𝑟 𝑖,𝑗 = (𝜃 𝑗 ) ⊤ 𝑥 𝑖 − 𝑦 𝑖,𝑗 𝜆 2 𝑗=1 𝑛 𝑢 𝑘=1 𝑛 𝜃 𝑘 𝑗 2
Given 𝜃 1 , 𝜃 2 , ⋯, 𝜃 𝑛 𝑢 , estimate 𝑥 1 , 𝑥 2 , ⋯, 𝑥 𝑛 𝑚
min 𝑥 (1) , 𝑥 (2) , ⋯, 𝑥 ( 𝑛 𝑚 ) 𝑖=1 𝑛 𝑚 𝑗:𝑟 𝑖,𝑗 = (𝜃 𝑗 ) ⊤ 𝑥 𝑖 − 𝑦 𝑖,𝑗 𝜆 2 𝑖=1 𝑛 𝑚 𝑘=1 𝑛 𝑥 𝑘 (𝑖) 2

29. Collaborative filtering optimization objective
Given 𝑥 1 , 𝑥 2 , ⋯, 𝑥 𝑛 𝑚 , estimate 𝜃 1 , 𝜃 2 , ⋯, 𝜃 𝑛 𝑢
min 𝜃 1 , 𝜃 2 , ⋯, 𝜃 𝑛 𝑢 𝑗=1 𝑛 𝑢 𝑖:𝑟 𝑖,𝑗 = (𝜃 𝑗 ) ⊤ 𝑥 𝑖 − 𝑦 𝑖,𝑗 𝜆 2 𝑗=1 𝑛 𝑢 𝑘=1 𝑛 𝜃 𝑘 𝑗 2
Given 𝜃 1 , 𝜃 2 , ⋯, 𝜃 𝑛 𝑢 , estimate 𝑥 1 , 𝑥 2 , ⋯, 𝑥 𝑛 𝑚
min 𝑥 (1) , 𝑥 (2) , ⋯, 𝑥 ( 𝑛 𝑚 ) 𝑖=1 𝑛 𝑚 𝑗:𝑟 𝑖,𝑗 = (𝜃 𝑗 ) ⊤ 𝑥 𝑖 − 𝑦 𝑖,𝑗 𝜆 2 𝑖=1 𝑛 𝑚 𝑘=1 𝑛 𝑥 𝑘 (𝑖) 2
Minimize 𝑥 1 , 𝑥 2 , ⋯, 𝑥 𝑛 𝑚 and 𝜃 1 , 𝜃 2 , ⋯, 𝜃 𝑛 𝑢 simultaneously
𝐽= 𝑗:𝑟 𝑖,𝑗 = (𝜃 𝑗 ) ⊤ 𝑥 𝑖 − 𝑦 𝑖,𝑗 𝜆 2 𝑗=1 𝑛 𝑢 𝑘=1 𝑛 𝜃 𝑘 𝑗 𝜆 2 𝑖=1 𝑛 𝑚 𝑘=1 𝑛 𝑥 𝑘 (𝑖) 2

30. Collaborative filtering optimization objective
𝐽( 𝑥 1 , 𝑥 2 , ⋯, 𝑥 𝑛 𝑚 , 𝜃 1 , 𝜃 2 , ⋯, 𝜃 𝑛 𝑢 )= 1 2 𝑟 𝑖,𝑗 =1 (𝜃 𝑗 ) ⊤ 𝑥 𝑖 − 𝑦 𝑖,𝑗 2 + 𝜆 2 𝑗=1 𝑛 𝑢 𝑘=1 𝑛 𝜃 𝑘 𝑗 2 + 𝜆 2 𝑖=1 𝑛 𝑚 𝑘=1 𝑛 𝑥 𝑘 (𝑖) 2

31. Collaborative filtering algorithm
Initialize 𝑥 1 , 𝑥 2 , ⋯, 𝑥 𝑛 𝑚 , 𝜃 1 , 𝜃 2 , ⋯, 𝜃 𝑛 𝑢 to small random values
Minimize 𝐽( 𝑥 1 , 𝑥 2 , ⋯, 𝑥 𝑛 𝑚 , 𝜃 1 , 𝜃 2 , ⋯, 𝜃 𝑛 𝑢 ) using gradient descent (or an advanced optimization algorithm). For every 𝑗= 1⋯ 𝑛 𝑢 , 𝑖=1, ⋯, 𝑛 𝑚 :
𝑥 𝑘 𝑗 ≔ 𝑥 𝑘 𝑗 −𝛼 𝑗:𝑟 𝑖,𝑗 =1 ( 𝜃 𝑗 ⊤ 𝑥 𝑖 − 𝑦 𝑖,𝑗 ) 𝜃 𝑘 𝑖 +𝜆 𝑥 𝑘 (𝑖)
𝜃 𝑘 𝑗 ≔ 𝜃 𝑘 𝑗 −𝛼 𝑖:𝑟 𝑖,𝑗 =1 ( 𝜃 𝑗 ⊤ 𝑥 𝑖 − 𝑦 𝑖,𝑗 ) 𝑥 𝑘 𝑖 +𝜆 𝜃 𝑘 (𝑗)
For a user with parameter 𝜃 and movie with (learned) feature 𝑥, predict a star rating of 𝜃 ⊤ 𝑥

32. Collaborative filtering
Movie
Alice (1)
Bob (2)
Carol (3)
Dave (4)
Love at last 5
Romance forever ?
Cute puppies of love 4
Nonstop car chases
Swords vs. karate

33. Collaborative filtering
Predicted ratings:
𝑋= − 𝑥 ⊤ − − 𝑥 ⊤ − ⋮ − 𝑥 𝑛 𝑚 ⊤ −
Θ= − 𝜃 ⊤ − − 𝜃 ⊤ − ⋮ − 𝜃 𝑛 𝑢 ⊤ −
Y=X Θ ⊤
Low-rank matrix factorization

34. Finding related movies/products
For each product 𝑖, we learn a feature vector 𝑥 (𝑖) ∈ 𝑅 𝑛
𝑥 1 : romance, 𝑥 2 : action, 𝑥 3 : comedy, …
How to find movie 𝑗 relate to movie 𝑖?
Small 𝑥 (𝑖) − 𝑥 (𝑗) movie j and I are “similar”

35. Recommender Systems Motivation Problem formulation
Content-based recommendations
Collaborative filtering
Mean normalization

36. Users who have not rated any movies
Alice (1)
Bob (2)
Carol (3)
Dave (4)
Eve (5)
Love at last 5 ?
Romance forever
Cute puppies of love 4
Nonstop car chases
Swords vs. karate
1 2 𝑟 𝑖,𝑗 = (𝜃 𝑗 ) ⊤ 𝑥 𝑖 − 𝑦 𝑖,𝑗 𝜆 2 𝑗=1 𝑛 𝑢 𝑘=1 𝑛 𝜃 𝑘 𝑗 𝜆 2 𝑖=1 𝑛 𝑚 𝑘=1 𝑛 𝑥 𝑘 (𝑖) 2
𝜃 (5) = 0 0

37. Users who have not rated any movies
Alice (1)
Bob (2)
Carol (3)
Dave (4)
Eve (5)
Love at last 5
Romance forever ?
Cute puppies of love 4
Nonstop car chases
Swords vs. karate
1 2 𝑟 𝑖,𝑗 = (𝜃 𝑗 ) ⊤ 𝑥 𝑖 − 𝑦 𝑖,𝑗 𝜆 2 𝑗=1 𝑛 𝑢 𝑘=1 𝑛 𝜃 𝑘 𝑗 𝜆 2 𝑖=1 𝑛 𝑚 𝑘=1 𝑛 𝑥 𝑘 (𝑖) 2
𝜃 (5) = 0 0

38. Mean normalization Learn 𝜃 (𝑗) , 𝑥 (𝑖) For user 𝑗, on movie 𝑖 predict:
𝜃 𝑗 ⊤ 𝑥 (𝑖) + 𝜇 𝑖
User 5 (Eve):
𝜃 5 = 𝜃 ⊤ 𝑥 (𝑖) + 𝜇 𝑖
Learn 𝜃 (𝑗) , 𝑥 (𝑖)

39. Recommender Systems Motivation Problem formulation
Content-based recommendations
Collaborative filtering
Mean normalization