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Matrix Factorization for Collaborative
Filtering is just Solving an Adjoint Latent
Dirichlet Allocation Model After All
Florian Wilhelm · Head of Data Science
Matrix Factorization
where
with set of users , items and latent dimension .
induces a personalized ranking .
2
X ⇡ X̂ := WHt
,
I
U
x̂ui = hwu, hii + bi
X 2 R|U|⇥|I|
, W 2 R|U|⇥|K|
, H 2 R|I|⇥|K|
K
>u
Research Questions
1. Why does matrix factorization work so well in
collaborative filtering tasks?
2. How can the factors and be interpreted?
3. What is the underlying data generating process?
3
W H
Interpret matrix factorization as a
Latent Dirichlet Allocation problem.
Classical Latent Dirichlet Allocation Model
4
|S|
|U|
|K|
Æ µu zus ius
Ø 'k
1. Choose
2. Choose
3. For user and interaction :
a) Choose cohort
b) Choose item
✓u ⇠ Dirichlet(↵).
'k ⇠ Dirichlet( ).
zus ⇠ Categorical(✓u).
ius ⇠ p(ius|'zus
) :=
Categorical('zus
).
u z
Shortcomings of Classical LDA for RecSys
1. Item preferences only depend on the user cohorts since
no explicit item popularity is included.
2. If existed, there would be no way of weighting the
item preferences of the cohort against the item
popularities for a user.
5
bi
bi
'k
bi
Matrix factorization does not
have those shortcomings.
LDA4Rec Model
6
Extends classical LDA with
item popularity and
user conformity .
Item probability:
i
u
ius ⇠ p(ius|'zus
, i, u) :=
Categorical(kck1
1
c)
c = 'zus
+ u ·
with
Reformulate MF as LDA4Rec
7
Sketch of Proof
8
Empirical Results on Movielens-100k
9
Conclusion
1. MF is equivalent to LDA4Rec, which is a plausible
model for the actual dynamics.
2. The factors and can be interpreted by
transforming them to the variables of LDA4Rec.
3. LDA4Rec gives us a data generating process for the
interaction matrix.
10
W H
Thank you!
Florian Wilhelm
Head of Data Science
inovex GmbH
Schanzenstraße 6-20
Kupferhütte 1.13
51063 Köln
florian.wilhelm@inovex.de

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Matrix Factorization for Collaborative Filtering Is Just Solving an Adjoint Latent Dirichlet Allocation Model After All

  • 1. Matrix Factorization for Collaborative Filtering is just Solving an Adjoint Latent Dirichlet Allocation Model After All Florian Wilhelm · Head of Data Science
  • 2. Matrix Factorization where with set of users , items and latent dimension . induces a personalized ranking . 2 X ⇡ X̂ := WHt , I U x̂ui = hwu, hii + bi X 2 R|U|⇥|I| , W 2 R|U|⇥|K| , H 2 R|I|⇥|K| K >u
  • 3. Research Questions 1. Why does matrix factorization work so well in collaborative filtering tasks? 2. How can the factors and be interpreted? 3. What is the underlying data generating process? 3 W H Interpret matrix factorization as a Latent Dirichlet Allocation problem.
  • 4. Classical Latent Dirichlet Allocation Model 4 |S| |U| |K| Æ µu zus ius Ø 'k 1. Choose 2. Choose 3. For user and interaction : a) Choose cohort b) Choose item ✓u ⇠ Dirichlet(↵). 'k ⇠ Dirichlet( ). zus ⇠ Categorical(✓u). ius ⇠ p(ius|'zus ) := Categorical('zus ). u z
  • 5. Shortcomings of Classical LDA for RecSys 1. Item preferences only depend on the user cohorts since no explicit item popularity is included. 2. If existed, there would be no way of weighting the item preferences of the cohort against the item popularities for a user. 5 bi bi 'k bi Matrix factorization does not have those shortcomings.
  • 6. LDA4Rec Model 6 Extends classical LDA with item popularity and user conformity . Item probability: i u ius ⇠ p(ius|'zus , i, u) := Categorical(kck1 1 c) c = 'zus + u · with
  • 7. Reformulate MF as LDA4Rec 7
  • 9. Empirical Results on Movielens-100k 9
  • 10. Conclusion 1. MF is equivalent to LDA4Rec, which is a plausible model for the actual dynamics. 2. The factors and can be interpreted by transforming them to the variables of LDA4Rec. 3. LDA4Rec gives us a data generating process for the interaction matrix. 10 W H
  • 11. Thank you! Florian Wilhelm Head of Data Science inovex GmbH Schanzenstraße 6-20 Kupferhütte 1.13 51063 Köln florian.wilhelm@inovex.de