This paper proposed an autoregressive model for rating prediction. The idea of this paper is similar to CF-RBM for collaborative filtering problem. Instead, it replaces an RMB with NADE. Since NADE is a mean-field approximation of RBM and also CF-RBM works for collaborative filtering problem, it makes sense that NADE model is applicable for CF problem.

The NADE model aims to estimate a probability of the given sequence as The previously seen sequence of rating can be used to predict the current rating. This also implies that some early rating on the known sequence has a strong influence on the current rating. This observation made NADE a different model from RNN.

The conditional probability distribution is defined as:

The model needs to estimate a score function such that it gives a high score when a user will likely to give a rating k to an item given her previous rating history.

The score function can be seen as a dot product of a user state with an item latent vector. . The bias is an item popularity and the dot product is the similarity between the current taste of the given user to the item .

The hidden state captures the user’s current interest. It means that this model captures the change of behavior. This hidden state is defined as The function g is an activation function, c is a bias, and a matrix W is an interaction matrix that transforms the input rating vector to a item vector.

The NADE-CF can be complicated but the main idea is the same as RBM-CF. Each user will have her own NADE network. NADE-CF shares all weight parameters among all users to avoid overfitting. Further, since each weight matrix associated with a rating, some weight matrix might be updated less frequently than others. The author changes the formula to compute the hidden state (look at the eq. 9 in the original paper). This way, the rare rating matrix will be updated more often.

In order to reduce the number parameters further, the low-rank matrix approximation is applied to matrix W and V. In addition, to improve the predictability further, the ordinal cost is introduced. The idea is if the user rating item m with rating k. Her rating preference of k-1 should be higher than k-2 and so on. In other words, if a user gave a rating of 3, it is likely that she will give a rating of 2 instead of 1 because a rating of 2 is closer to a rating of 3. The ordinal cost then is modeled as a rank loss similar to the list-wise learning to rank model.

Overall, NADE-CF yields a very good performance on the popular CF data set such as Netflix and MovieLens. The model is complicated and seems to be slow. However, with ordinal cost, low-rank approximation, and its ability to have a deeper model, made this model more attractive and it might be worthwhile to investigate an autoregressive model for this particular task.