pyalswr 1.0.0

UNKNOWN

# ALS-WR : Alternating-Least-Squares with Weighted-λ-Regularization

This is the code to perform ALS-WR as presented by [Zhou et al]

The code is mainly inspired from [Mendeley Ltd]

- *R* ∈ ℝ^{*n* × *p*} = (*r*_*i**j*) denote the sparse user-item matrix.
- 𝒟 is the set of available ratings.
- Assume a low rank constraint *K* on the matrix : *R* = *U**V*^*T*, *U*, *V* ∈ ℝ^{*n* × *K*}, ℝ^{*p* × *K*}
- *U*_*i* and *V*_*j* denotes respectively the latent features of user *i* and item *j*.
- *n*_{*u*_*i*} is the number of ratings provided by user *i*.
- *n*_{*v*_*j*} is the number of ratings available for item *j*.
- *J*_*i* and *L*_*j* is respectively the set of rated item by user *i* and available ratings for item *j*.

Objective function :
*f*(*U*, *V*)=∑_{*i*, *j* ∈ 𝒟}(*r*_*i**j* − *U*_*i**V*_*j*^*T*)² + *λ*(∑_*i**n*_{*u*_*i*}||*U*_*i*||²+∑_*i**n*_{*v*_*j*}||*V*_*j*||²)

**Algorithm**

- Input : *R*, *K* (possibly an initialization of *U* or *V* with SVD)
- Until convergence do
- For each user (in parallel if wanted)
- *U*_*i* = (*V*_{*I*_*i*}*V*_{*I*_*i*} + *λ**I*_*K*)⁻¹*V*_{*I*_*i*}*R*_{*i*, *I*_*i*}
- For each item (in parallel if wanted)
- *V*_*j* = (*U*_{*L*_*j*}*U*_{*L*_*j*} + *λ**I*_*K*)⁻¹*U*_{*L*_*j*}*R*_{*L*_*j*, *j*}

[Zhou et al]: http://www.grappa.univ-lille3.fr/~mary/cours/stats/centrale/reco/paper/MatrixFactorizationALS.pdf
[Mendeley Ltd]: https://github.com/Mendeley/mrec/blob/master/mrec/mf/wrmf.py