Input Data

Our input data, for collaborative filtering, consists of:

U

The set of users in the system or data set.

I

The set of items in the system or data set.

R

The set of ratings in the data set. R can be used as either a set of rating observations or as a (partially observed) |U| \times |I| matrix; context makes it clear which meaning is intended.

Within each of these sets, we can refer to individual entries:

u, v \in U

I use u and v as variables referencing individual users. If I need more than two, then I use numeric subscripts u_1, u_2, etc.

i, j \in I

I use i and j as variables referencing individual items. Again, for more than two, I use numeric subscripts. This does mean that i is not available as a counter variable, but I do not find that to be too much of a difficulty in practice.

r_{ui} \in R

An individual rating value, the rating user u gave for item i. In an implicit feedback setting, this takes on whatever value you are using as the ‘rating’: 1/0, a play count, etc.

One advantage of always using u,v for users and i,j for items is that meaning is clear from looking at a variable or subscripted variable. It also allows the following subset notations:

I_u \subset I

The set of items rated by user u.

U_i \subset U

The set of users who have rated or purchased item i.

R_u \subset R

The set of ratings given by user u

R_i \subset R

The set of items for item i

\vec{r}_u

User u’s rating vector, an |I|-dimensional vector with missing values for unrated items.

\vec{r}_i

Item i’s rating vector, a |U|-dimensional vector with missing values for users who have not rated the item.

Scores and Similarities

With this notation, we can write things like the user-user rating prediction formula:

\hat r_{ui} = s(i;u) = \frac{\sum_{v \in N(u,i)} w(u,v)(r_{vi}-\bar r_v)}{\sum_{v \in N(u,i)} |w(u,v)|} + \bar r_u

N(u;i) is the neighborhood for user u for the purpose of scoring item i, and \bar r_u is user u’s average rating. I explicitly state the ranges of my summations for two reasons, even if they are implicitly clear for experienced recsys researchers: it is a common tripping point for students, and it is a place where subtle implementation differences get buried. I find that it is not overly cumbersome with this notation.

Matrix Factorization

For basic decomposition of the ratings matrix, I usually use P and Q these days:

R \approx PQ^{\mathrm{T}}

We then have user vectors \vec p_u and item vectors \vec q_i. I like using the transpose notation for the factorization, so that latent features are on the columns of both P and Q.

Advanced decompositions that involve more than two matrices will need to find additional matrices; one unfortunate tradeoff of using U for the set of users is that it is no longer available for a matrix.