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"description": "# ALS-WR : Alternating-Least-Squares with Weighted-\u03bb-Regularization \n\nThis is the code to perform ALS-WR as presented by [Zhou et al] \n\nThe code is mainly inspired from [Mendeley Ltd]\n\n- *R*\u2004\u2208\u2004\u211d*n*\u2005\u00d7\u2005*p*\u2004=\u2004(*r**i**j*) denote the sparse user-item matrix.\n- \ud835\udc9f is the set of available ratings.\n- Assume a low rank constraint *K* on the matrix : *R*\u2004=\u2004*U**V**T*, *U*,\u2006*V*\u2004\u2208\u2004\u211d*n*\u2005\u00d7\u2005*K*,\u2006\u211d*p*\u2005\u00d7\u2005*K*\n- *U**i* and *V**j* denotes respectively the latent features of user *i* and item *j*.\n- *n**u**i* is the number of ratings provided by user *i*.\n- *n**v**j* is the number of ratings available for item *j*.\n- *J**i* and *L**j* is respectively the set of rated item by user *i* and available ratings for item *j*.\n\nObjective function :\n*f*(*U*,\u2006*V*)=\u2211*i*,\u2006*j*\u2004\u2208\u2004\ud835\udc9f(*r**i**j*\u2005\u2212\u2005*U**i**V**j**T*)2\u2005+\u2005*\u03bb*(\u2211*i**n**u**i*||*U**i*||2+\u2211*i**n**v**j*||*V**j*||2)\n\n **Algorithm**\n\n- Input : *R*, *K* (possibly an initialization of *U* or *V* with SVD)\n- Until convergence do\n - For each user (in parallel if wanted)\n - *U**i*\u2004=\u2004(*V**I**i**V**I**i*\u2005+\u2005*\u03bb**I**K*)\u22121*V**I**i**R**i*,\u2006*I**i*\n - For each item (in parallel if wanted)\n - *V**j*\u2004=\u2004(*U**L**j**U**L**j*\u2005+\u2005*\u03bb**I**K*)\u22121*U**L**j**R**L**j*,\u2006*j*\n\n [Zhou et al]: http://www.grappa.univ-lille3.fr/~mary/cours/stats/centrale/reco/paper/MatrixFactorizationALS.pdf\n [Mendeley Ltd]: https://github.com/Mendeley/mrec/blob/master/mrec/mf/wrmf.py\n",
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