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Matrix factorization for recsys — objective.

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Answer

  • Model rating ruir_{\mathrm{ui}} ≈ pup_{u}' * qiq_{i} where pup_{u} ∈ RkR^{k} is user vector, qiq_{i} is item vector.
  • Minimize Σ (rui  −  puqi)2  +  λ(r_{\mathrm{ui}}\; - \;p_{u}q_{i})^{2}\; + \;{\lambda} (∣∣pu∣∣2  +  ∣∣qi∣∣2)( \mid \mid p_{u} \mid \mid ^{2}\; + \; \mid \mid q_{i} \mid \mid ^{2}) over observed entries.
  • Solved with SGD or ALS.
  • Handles missing entries naturally (only sum over observed).
  • Extensions: biases (puqi  +  bu  +  bi  +  μ)(p_{u}q_{i}\; + \;b_{u}\; + \;b_{i}\; + \;{\mu}), implicit feedback (BPR loss, weighted ALS in Spark).
Check yourself — multiple choice
  • Random
  • ruir_{\mathrm{ui}} ≈ pup_{u}'qiq_{i} minimizing squared error + L2 regularization over observed entries; SGD or ALS; extended with biases + implicit feedback (BPR)
  • Same as OLS
  • Not real

Recsys MF: ruir_{\mathrm{ui}} ≈ pup_{u}'qiq_{i} via SGD/ALS; extensions for bias / implicit / BPR.

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