# Notation standards, References¶

In the documentation, you will find the following notation:

• $$R$$ : the set of all ratings.

• $$R_{train}$$, $$R_{test}$$ and $$\hat{R}$$ denote the training set, the test set, and the set of predicted ratings.

• $$U$$ : the set of all users. $$u$$ and $$v$$ denotes users.

• $$I$$ : the set of all items. $$i$$ and $$j$$ denotes items.

• $$U_i$$ : the set of all users that have rated item $$i$$.

• $$U_{ij}$$ : the set of all users that have rated both items $$i$$ and $$j$$.

• $$I_u$$ : the set of all items rated by user $$u$$.

• $$I_{uv}$$ : the set of all items rated by both users $$u$$ and $$v$$.

• $$r_{ui}$$ : the true rating of user $$u$$ for item $$i$$.

• $$\hat{r}_{ui}$$ : the estimated rating of user $$u$$ for item $$i$$.

• $$b_{ui}$$ : the baseline rating of user $$u$$ for item $$i$$.

• $$\mu$$ : the mean of all ratings.

• $$\mu_u$$ : the mean of all ratings given by user $$u$$.

• $$\mu_i$$ : the mean of all ratings given to item $$i$$.

• $$\sigma_u$$ : the standard deviation of all ratings given by user $$u$$.

• $$\sigma_i$$ : the standard deviation of all ratings given to item $$i$$.

• $$N_i^k(u)$$ : the $$k$$ nearest neighbors of user $$u$$ that have rated item $$i$$. This set is computed using a similarity metric.

• $$N_u^k(i)$$ : the $$k$$ nearest neighbors of item $$i$$ that are rated by user $$u$$. This set is computed using a similarity metric.

References

Here are the papers used as references in the documentation. Links to pdf files where added when possible. A simple Google search should lead you easily to the missing ones :)

GM05

Thomas George and Srujana Merugu. A scalable collaborative filtering framework based on co-clustering. 2005. URL: https://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.113.6458&rep=rep1&type=pdf.

Kor08

Yehuda Koren. Factorization meets the neighborhood: a multifaceted collaborative filtering model. 2008. URL: https://people.engr.tamu.edu/huangrh/Spring16/papers_course/matrix_factorization.pdf.

Kor10

Yehuda Koren. Factor in the neighbors: scalable and accurate collaborative filtering. 2010. URL: https://courses.ischool.berkeley.edu/i290-dm/s11/SECURE/a1-koren.pdf.

KBV09

Yehuda Koren, Robert Bell, and Chris Volinsky. Matrix factorization techniques for recommender systems. 2009.

LS01

Daniel D. Lee and H. Sebastian Seung. Algorithms for non-negative matrix factorization. 2001. URL: https://papers.nips.cc/paper/1861-algorithms-for-non-negative-matrix-factorization.pdf.

LM07

Daniel Lemire and Anna Maclachlan. Slope one predictors for online rating-based collaborative filtering. 2007. URL: https://arxiv.org/abs/cs/0702144.

LZXZ14

Xin Luo, Mengchu Zhou, Yunni Xia, and Qinsheng Zhu. An efficient non-negative matrix factorization-based approach to collaborative filtering for recommender systems. 2014.

RRSK10

Francesco Ricci, Lior Rokach, Bracha Shapira, and Paul B. Kantor. Recommender Systems Handbook. 1st edition, 2010.

SM08

Ruslan Salakhutdinov and Andriy Mnih. Probabilistic matrix factorization. 2008. URL: https://papers.nips.cc/paper/3208-probabilistic-matrix-factorization.pdf.

ZWFM96

Sheng Zhang, Weihong Wang, James Ford, and Fillia Makedon. Learning from incomplete ratings using non-negative matrix factorization. 1996. URL: https://www.siam.org/meetings/sdm06/proceedings/059zhangs2.pdf.