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

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

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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

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

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

  1. 1. Matrix Factorization for Collaborative Filtering is just Solving an Adjoint Latent Dirichlet Allocation Model After All Florian Wilhelm · Head of Data Science
  2. 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. 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. 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. 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. 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. 7. Reformulate MF as LDA4Rec 7
  8. 8. Sketch of Proof 8
  9. 9. Empirical Results on Movielens-100k 9
  10. 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. 11. 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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