T H E U V D E C O M P O S I T I O N A L G O R I T H M

Motivation


It is now common to get “personal recommendations” when we visit a website.

News articles


Product recommendations
Advertisements
Why ?


Unlike paper newspapers or brick and mortar stores, there is no limit [in terms of space/inventory] what can be shown or sold on a web-site..
Long-Tail effect

Large part of the income comes from the tail [example in search revenue]

The Netflix challenge




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Netflix is a (online) US company from where people can rent movies
Netflix would like to recommend movies to users.
Netflix challenge (2006) – one million dollars prize who could beat their movie recommendation by 10%
After three years the prize was awarded..
We will discuss one of the “main ideas” behind the winning entry
 [We will follow the discussion from the “Mining of
Massive Data Sets”]

The Data
 Data can be arranged in the form…
M1 M2 M3 M4 M5 M6 M7
U1
U2 2
3
2
2
4
5
U3
U4
2
3
1 3
2
4
4
4
Users have given rating (between 1 and 5) from a database of 7 movie
..
What should be recommended to U2 ?

First ideas
 Do missing entries mean a rating of “0” ?
 How about simple dot product.
 Other ideas…?

Clustering

Association Rules ?

Another basic idea
 Let the user “u” rate movie “m” as follow:
 Take the average of the following two numbers.


Average of user’s “u” ratings.
Average of all ratings given to movie “m” by all users’ who rated “m”
 This was only 3% worse than than the Netflix algorithm (called CineMatch)

Latent Modeling
 A big breakthrough is the idea of “latent variable modeling”


The data we observe is a result of another variable or set of variables which are “latent” (not observable)…
The latent variables generate the observed data…

In our case…the latent variables control the generation of the ratings..
 So the challenge is to “infer” the latent variables from the observed data…

 clustering
Bayes Theorem

Cognitive Science/AI
 Scientists working in AI/Cognitive Science have drawn the following analogy..

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Mind  Computer
Mental Representation  Programs/Theories
Thinking  Computational Process/Algorithms
 Practical Outcome: Infer Latent Structures

The Key Idea

Decompose the User x Rating matrix into:

User x Rating = ( User x Genre ) x (Genre x Movies)
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Number of Genres is typically small
 Or

R =~ UV

Find U and V such that ||R – UV|| is minimized…
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Almost like k-means clustering…why ?

Example of UV Decomposition
The criterion used to select U and V is the Root Mean Square Error (RMSE)

RMSE Example
1 2
3
1 3
2 4
1
3
(1
-
1)
2 +
(2
-
3)
2 +
(3
-
4)
2 =
1
3
0
+
1
+
1
=
1
3
2

Example..Continued
Results in an RMSE of 1.8

UV Computation

UV Computation..continued
Same calculation as before…..

UV Computation….

UV Decomposition
 The above process can be generalized to any entry…
 Continue the process until RMSE settles into a local optimal…
 So in spirit very similar to k-means..

References
 Mining of Massive Data Sets

Rajaraman, Leskovic, Ullman