Paper:
Coevolutionary Latent Feature Processes for Continuous-Time User-Item Interactions
Matching users to the right items at the right time is a fundamental task in recommendation systems. As users interact with different items over time, users' and items' feature may evolve and co-evolve over time. Traditional models based on static latent features or discretizing time into epochs can become ineffective for capturing the fine-grained temporal dynamics in the user-item interactions. We propose a coevolutionary latent feature process model that accurately captures the coevolving nature of users' and items' feature. To learn parameters, we design an efficient convex optimization algorithm with a novel low rank space sharing constraints. Extensive experiments on diverse real-world datasets demonstrate significant improvements in user behavior prediction compared to state-of-the-arts.

Review:
The paper introduces a model to predict the time of interactions between users and items, and the item which is the most promising for a user. This model builds upon a feature representation of users and items which shifts along time and is a weigthed sum of (i) past actions, (ii) past external features, and (iii) representations of users/items which did interact in the past. The law of time between interactions is represented by a Hawkes process based on the dot product between users and items representations.  The model is slightly changed to allow for a loss function (negative log likelihood) convex in the parameters. Experiments show that the proposed model predicts the interaction time with a much better accuracy than previous similar model (LowRankHawkes) while it similarly predicts the item to interact with. 
Modeling the evolution of tastes/properties of users/items along time is a challenging problem for Machine Learning. This paper proposes a solution to model that evolution in a continuous manner, without relying on cyclicity or constant shift assumptions. Experimental results support the proposed model.  The paper was a pleasure to read. It puts apart technical points to focus on what is needed to understand the approach and use it. As an example, Section 4.2 recalls the conditional gradient algorithm but does not give closed forms of the gradients.        Still, I was surprised/disappointed by three points:  1) Is matrix eta necessary ? This matrix contradicts the fact that a user/item evolve along time. Did you measure by any way the necessity for eta ? By example you could look at the Frobenius norm of eta compared to the Frobenius norm of $\Sum_ix_iX_i$, or indicates prediction results of the same model without eta.  2) Theorem 1 is not a Theorem. Theoretical results on nuclear norm minimization to recover rank minimization require assumptions on the matrices and on the observed entries. Such assumptions are not listed in current paper. The authors only give a sketch proof on how to support the proposed reparameterization.  3) Conclusions of Theorem 1 are excessive. Under some hypothesis, we probably can prove that the algorithm learns a matrix X of (low-)rank $k$, but that rank will be greater than the rank of A, B, C and D. Thereafter, X_1, X_5, X_6 and X_7 can live in different column spaces.     To increase the practical usability of the proposed model, I'm looking forward extensions of that work (i) considering incremental update of the model to manage the never-ending flow of data, (ii) managing the exploration/exploitation tradeoff while using that model to recommend interactions to users.        ****** typos ******  Figure 1(a): at the bottom, isn't it $I_j(t_k^***u***)$ as it feeds user $u$ representation (if so, correct also the caption).  Figure 1(a): Arrows are a bit messy. I would expect the arrow from $U_u(t_1^u)$ to go to $I_i(t_l^u)$ as the effect of the interaction is taken into account in the future. Similarly, I would not expect any arrow from $q_j^{u,i}$ to $I_i(t_1^u)$; there is already an arrow from $q_j^{u,i}$ to $I_i(t_l^u)$. If these remarks make sens, there is also two arrows at time $t_l^u$ which should disappear.  Figure 1(b): I would expect an arrow from node $j$ to represent the fact that the representation of that item is poorly aligned with the representation of user $u$.  L.161: "[...] item $i$ at time ***$t$*** than [...]"  L.162: "[...] item $i$ than item ***$j$***. [...]"