Performance metric in recommender systems with implicit feedback This paper describes a technique for making recommendations when the feedback is implicit, that is, $r_{ui}$ is only a guess. The recommendation problems boils down to the following optimisation problem:
$$
\min_{x^*, y^*}\ \ \sum_{u,i}c_{ui}(p_{ui}-x_i^Ty_i)^2+\lambda \left(\sum_u\|x_u\|^2+\|y_u\|^2 \right) 
$$
where $c_{ui} = 1+\alpha r_{ui}$.   

I have the following concerns:


The algorithm above approximates the binary indicator function $p_{ui}$, weighted by the term $c_{ui}$. However, most people use RMSE between $r_{ui}$ and $x_i^Ty_i$ for assessing performance. I am not sure why that would make sense. 
Why using RMSE at all? With implicit feedback, it would make more sense to me to use rank-based metrics. I would imagine RMSE is more suitable in machine learning competition where the objective metric is usually RMSE.
I think your post is missing a clear question. 

I'll answer your concerns anyway:


For a implicit feedback problem like the one described RMSE, indeed, doesn't make much sense. First of all it is not clear what the difference between the real value (number of interactions, e.g. watched movie 4 times) and the estimation would mean. Second, in recommender systems you usuallly want to give the user 10-20 relevant recommendations and you don't care for the rest. Hence, it makes sense to optimize your algorithm to get these top recommendations right while disregarding your performance further down in the ranking. 
See my first answer. Also, the authors of your cited paper argue for a rank-based metric.