Collaborative filtering(recommendation) relies on groups with similar properties or preferences to the current user.\\
The data scope for collaborative filtering is comprised of users who have expressed preferences on Items, action referenced from this point as rating, using the following triple representation User,Item,Rating.\\
Other types of data can be also represented, as it’s done by large ecommerce sites: hasPurchased, hasViewed.. These are called unary ratings.\\
The whole set of ratings triples is represented as a tuple User Item.
In order to infer the relevance for a user a sparse matrix with the tuples User,Item is created.

There are different types of algorithms used to provide these recommendations:\\
\textbf{Baseline predictors}\\
These are methods used before any type of personalized /sophisticated algorithm are deployed. They do not depend on user’s ratings and can be of use while providing predictions for new users.\\
For a user u  and item I, the prediction is given by the formula:\\
$B_{ui}=\mu$, where u is the average overall rating
The general formula for a baseline predictor is given by
$b_{ui}=\mu+b_u+b_i$\\\\
This sort of algorithms can be further tuned by computing other baselines that can respond to various effects ( a new item which is added to the site with no ratings..). They mostly capture effects of user bias, item popularity and can be leveraged over a large timeframe.\\\\
\textbf{User-User collaborative filtering}
This is one of the first automated collaborative filtering methods. At its core is a direct implementation of the core premise of collaborative filtering: based on other users with similar preferences, use their rating to predict what the current user will like.\\
The likeliness of a user liking an item is calculated using a weighted average of neighboring user’s ratings using the similarity as weight.\\
Using u as the user and I as the item, $s(u,u')$ as the similarity between the users
$r_{ui,i}$ as the rating for item i by user $u_i$
$r_{ui}$ as the average rating (to compensate for users who give generally higher scores\\
The similarity computation is analogous to the one in the next section.