# What is $r_u{_v}$ in the kNN algorithm for collaborative filtering?

$\hat{r}_u{_i} = \frac{\Sigma_j{_∈}{_N}{_{_k}}{_(}{_u}{_,}{_i}{_)} > c_i{_j} r_u{_v}} > {\Sigma_j{_∈}{_N}{_{_k}}{_(}{_u}{_,}{_i}{_)} c_i{_j}}$

where $N_k{_(}{_u}{_,}{_i}{_)}$ denotes the set of the k items most similar to item i that were rated by user u.

what is $r_u{_v}$? The user rating for item v, but what is v?

The meaning of the indicies of $r$ is spelled out on the first page of the paper
We define the set of users as the set of integers {1, . . . , m} and the set of items as the set of integers {1, . . . , n}. The $m \times n$ rating matrix $R = [r_{ui}] \ 1 \leq u \leq m;1 \leq i \leq n$ stores the rating sof users for items where $r_{ui}$ is the rating of user u for item i.
So $v$ is one of the items in the suite of objects you are attempting to recommend. Or, more pedantically, the index of one of the items after you have assigned unique indices to every item in your suite.
In the equation (1), I believe the $v$ is a type left over from a copy and paste. It should be a $j$, i.e. we are summing over all the similar items and weighting their rating by the similarity.
• I don't think this answers the question. Which item is $v$ actually referring to? – RoryT Jul 25 '17 at 3:53