Matrix factorization-based methods are among the most popular methods for collaborative filtering tasks with implicit feedback. The most effective of these methods do not apply sign constraints, such as non-negativity, to their factors. Despite their simplicity, the latent factors for users and items lack interpretability, which is becoming an increasingly important requirement. In this work, we provide a theoretical link between unconstrained and the interpretable non-negative matrix factorization in terms of the personalized ranking induced by these methods. We also introduce a novel, latent Dirichlet allocation-inspired model for recommenders and extend our theoretical link to also allow the interpretation of an unconstrained matrix factorization as an adjoint formulation of our new model. Our experiments indicate that this novel approach represents the unknown processes of implicit user-item interactions in the real world much better than unconstrained matrix factorization while being interpretable. This talk was presented at 15th ACM Conference on Recommender Systems in Amsterdam (RecSys 2021). Find more information under https://dl.acm.org/doi/fullHtml/10.1145/3460231.3474266

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

8. Sketch of Proof
8

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