> For the complete documentation index, see [llms.txt](https://amanalok.gitbook.io/recommender-systems/llms.txt). Markdown versions of documentation pages are available by appending `.md` to page URLs; this page is available as [Markdown](https://amanalok.gitbook.io/recommender-systems/collaborative-filtering.md).

# Collaborative Filtering

Consider a list of movies as "items". Each movie has an associated rating which we will call "score". With this, we write the following denotion:

$$
s(j) = score \hspace{0.1cm} for \hspace{0.1cm} item \hspace{0.1cm} j
\\
where, \hspace{0.1cm} j = 1 \hspace{0.1cm} ... \hspace{0.1cm} M, \hspace{0.1cm} if  \hspace{0.1cm} there \hspace{0.1cm} are \hspace{0.1cm} M \hspace{0.1cm} items.
$$

​Basic algorithm is to make average rating for item $$j$$​:

$$
s(j) = \frac{\sum\_{i\isin\Omega\_j} r\_{i,j}}{|\Omega\_j|}
\\
where, \Omega\_j \hspace{0.1cm} \equiv set \hspace{0.1cm} of \hspace{0.1cm} all \hspace{0.1cm} users \hspace{0.1cm} who \hspace{0.1cm} rated \hspace{0.1cm} item \hspace{0.1cm} j
\\
r\_{i,j} \equiv rating, \hspace{0.1cm} user \hspace{0.1cm} i \hspace{0.1cm} gave \hspace{0.1cm} item \hspace{0.1cm} j
$$

### ​Score Personalisation

Score $$s(i, j)$$ can depend on both user $$i$$​ and item (e.g. movie) $$j$$.

$$
s(i,j) = \frac{\sum\_{i^` \isin \Omega_j} r_{i^`j}}{|\Omega\_j|}
\\
where  \hspace{0.1cm} i = 1...N, \hspace{0.5cm} N \equiv No. \hspace{0.1cm} of \hspace{0.1cm} users
$$

The above equation does not change anything from the equation to calculate the average rating as  the score still does not depend on user $$i$$​. So, every user still sees the same score for each item.

The above equation is actually meant to introduce a new convention and symbols which we would modify later on to introduce personalisation. We would observe this in an upcoming blog article.

### Ratings Matrix

The central object for any recommender systems is the ratings matrix. It is defined as follows:

$$
r\_{i,j} = rating \hspace{0.1cm} user \hspace{0.1cm} I \hspace{0.1cm} gave \hspace{0.1cm} item \hspace{0.1cm} j
\\
where \hspace{0.1cm} i = 1...N, \hspace{0.1cm} j = 1...M
\\
R\_{N \times M} \equiv user-item \hspace{0.1cm} ratings \hspace{0.1cm} matrix \hspace{0.1cm} of \hspace{0.1cm} size \hspace{0.1cm} N \times M
$$

#### Sparsity

User-item ratings matrix is sparse because most entries are generally empty. Below is an example of a dummy ratings matrix. Each row represents a user and each column represents an item (e.g. movie).

$$
\begin{bmatrix}

* & 4.5 & 2.0 & \*\\
  4.0 & \* & 3.5 & \*\\
* & 5.0 & \* & 2.0\\
* & 3.5 & 4.0 & 1.0
  \end{bmatrix}
  $$

$$\*$$ marked entries in the above matrix mean, there are no ratings (score) for the specific user-item pair.

### Collaborative Filtering Goal

We want to make recommendations. Thus, its a good thing that the ratings matrix generally is sparse, in order to actually have items to recommend.

We want to guess what a user might rate a movie, the user has not seen yet. Thus,

$$
s(i,j) \equiv ^{r}(i,j) \equiv guess \hspace{0.1cm} what \hspace{0.1cm} user \hspace{0.1cm} i \hspace{0.1cm} might \hspace{0.1cm} rate \hspace{0.1cm} item \hspace{0.1cm} j
$$

​We want to predict a real number, so objective of this task would be "mean squared error" (MSE).

$$
MSE = \frac{1}{|\Omega|} \sum\_{i,j \isin \Omega} (r\_{ij} - ^{r}\_{ij})^2
\\
where, \Omega \equiv Set \hspace{0.1cm} of \hspace{0.1cm} pairs \hspace{0.1cm} (i,j) \hspace{0.1cm} where \hspace{0.1cm} user \hspace{0.1cm} i \hspace{0.1cm} has \hspace{0.1cm} rated \hspace{0.1cm} item \hspace{0.1cm} j.
$$

### Collborative Filtering Types

There are 2 different ways to apply collaborative filtering for making recommendations:

* User-User Collaborative Filtering
* Item-Item Collaborative Filtering