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Project Description

# 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<sup>*n* × *p*</sup> = (*r*<sub>*i**j*</sub>) 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*<sup>*T*</sup>, *U*, *V* ∈ ℝ<sup>*n* × *K*</sup>, ℝ<sup>*p* × *K*</sup>
- *U*<sub>*i*</sub> and *V*<sub>*j*</sub> denotes respectively the latent features of user *i* and item *j*.
- *n*<sub>*u*<sub>*i*</sub></sub> is the number of ratings provided by user *i*.
- *n*<sub>*v*<sub>*j*</sub></sub> is the number of ratings available for item *j*.
- *J*<sub>*i*</sub> and *L*<sub>*j*</sub> is respectively the set of rated item by user *i* and available ratings for item *j*.

Objective function :
*f*(*U*, *V*)=∑<sub>*i*, *j* ∈ 𝒟</sub>(*r*<sub>*i**j*</sub> − *U*<sub>*i*</sub>*V*<sub>*j*</sub><sup>*T*</sup>)<sup>2</sup> + *λ*(∑<sub>*i*</sub>*n*<sub>*u*<sub>*i*</sub></sub>||*U*<sub>*i*</sub>||<sup>2</sup>+∑<sub>*i*</sub>*n*<sub>*v*<sub>*j*</sub></sub>||*V*<sub>*j*</sub>||<sup>2</sup>)


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

[Zhou et al]:
[Mendeley Ltd]:

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Filename, Size & Hash SHA256 Hash Help File Type Python Version Upload Date
(3.0 kB) Copy SHA256 Hash SHA256
Wheel py2.py3 Mar 26, 2016
(2.5 kB) Copy SHA256 Hash SHA256
Source None Mar 26, 2016

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