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- 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