Collaborative filtering through matrix factorization with logistic loss function Consider collaborative filtering problem. We have matrix $M$ of size #users * #items. $M_{i,j} = 1$ if user i likes item j, $M_{i,j} = 0$ if user i dislikes item j and $M_{i,j}=?$ if there is no data about (i,j) pair. We want to predict $M_{i,j}$ for future user, item pairs.
Standard collaborative filtering approach is to represent M as product of 2 matrices $U \times V$ such that $||M - U \times V||_2$ is minimal (e.g. minimizing mean square error for known elements of $M$).
To me logistic loss function seems more suitable, why are all algorithms using MSE?
 A: Most of the papers you'll find on the subject will deal with matrices where the ratings are on a scale [0,5]. In the context of the Netflix Prize for example, matrices have discrete ratings from 1 to 5 (+ the missing values). That's why the squared error is the most spread cost function. Some other error measures such as the Kullback-Leibler divergence can be seen.
Another problem that can occur with standard matrix factorization is that some of the elements of the matrices U and V may be negative (particularly during the first steps). That's a reason why you wouldn't use the log-loss here as your cost function.  
However, if you're talking about Non-negative Matrix Factorization you should be able to use the log-loss as your cost function. You are in a  similar case than Logistic Regression where log-loss is used as the cost function: your observed values are 0's and 1's and you predict a number (probability) between 0 and 1.  
A: We use logistic loss for implicit matrix factorization at Spotify in the context of music recommendations (using play counts). We've just published a paper on our method in an upcoming NIPS 2014 workshop. The paper is titled Logistic Matrix Factorization for Implicit Feedback Data and can be found here http://stanford.edu/~rezab/nips2014workshop/submits/logmat.pdf
Code for the paper can be found on my Github https://github.com/MrChrisJohnson/logistic-mf
