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Title: Interpretable Robust Recommender Systems with Side Information

Paper content:
In this paper, we propose two methods, namely Trace-norm regression (TNR) and Stable Trace-norm Analysis (StaTNA), to improve performances of recommender systems with side information.
Our trace-norm regression approach extracts low-rank latent factors underlying the side information that drives user preference under different context.
Furthermore, our novel recommender framework StaTNA not only captures latent low-rank common drivers for user preferences, but also considers idiosyncratic taste for individual users.
We compare performances of TNR and StaTNA on the MovieLens datasets against state-of-the-art models, and demonstrate that StaTNA and TNR in general outperforms these methods.
The boom of user activity on e-commerce and social networks has continuously fueled the development of recommender systems to most effectively provide suggestions for items that may potentially match user interest.
In highlyrated Internet sites such as Amazon.com, YouTube, Netflix, Spotify, LinkedIn, Facebook, Tripadvisor, Last.fm, and IMDb, developing and deploying personalized recommender systems lie at the crux of the services they provide to users and subscribers (Ricci et al., 2015) .
For example, Youtube, one of the worlds most popular video sites, has deployed a recommender system that updates regularly to deliver personalized sets of videos to users based on their previous or recent activity on site to help users find videos relevant to their interests, potentially keeping users entertained and engaged BID5 .Among
the vast advancements in deep learning and matrix completion techniques to build recommender systems (Ricci BID21 , one of the most imperative aspect of research in such area is to identify latent (possibly low-rank) commonalities that drive specific types of user behaviour. For example
, BID6 proposes a deep neural network based matrix factorization approach that uses explicit rating as well as implicit ratings to map user and items into common low-dimensional space. Yet, such
variety of low-rank methodologies do not address the impact of idiosyncratic behaviour among buyers, which may potentially skew the overall learned commonalities across user groups.In this work, we propose two multi-task learning methods to improve performances of recommender systems using contextual side information. We first
introduce an approach based on trace-norm regression (TNR) that enables us to extract low-rank latent dimensions underlying the side information that drive user preference according to variations in context, such as item features, user characteristics, time, season, location, etc. This is
achieved by introducing a nuclear-norm regularization penalty term in the multi-task regression model, and we highlight that such latent dimensions can be thought of as homogeneous behaviour among particular types of user groups. Furthermore
, we propose a novel recommender framework called Stable Trace-norm Analysis (StaTNA) that not only captures latent low-rank common drivers for user preference, but also considers idiosyncratic taste for individual users. This is achieved
by, in addition to the low-rank penalty, adding a sparsity regularization term to exploit the sparse nature of heterogeneous behaviour. Finally, we test
the performance of StaTNA on the MovieLens datasets against state-of-the-art models, and demonstrate that StaTNA and TNR in general outperforms these methods.
As mentioned in earlier sections, we are interested in analyzing particular underlying commonalities in user preferences.
We achieve this by investigating the principal components of our estimate of the low-rank matrix L, each of which we consider as a common type of user preference.
Since our estimated L is of rank 6, we conclude that there are 6 major common types of user preferences, whose component scores (i.e. explained variance percentages) are listed in Table 4 , where we observe that the first principal component explains 88.94% of the variability in user ratings.
Table 5 .
Top 12 features of highest absolute weights within the first two principal components (PC1 and PC2).
Details of other principle components are shown in TAB9 in Appendinx C.2.
Our methodology to solve TNR and StaTNA (i.e. Algorithm 1 in Appendix A.1) may be computationally expensive when the matrix is large since it requires calling a Singular Value Decomposition (SVD) oracle in each iteration of the algorithm.
Hence we propose two alternative methods, a FW-T algorithm and a nonconvex reformulation of the problem, to avoid using an SVD oracle.
These are detailed in Appendix A.2.
Furthermore, our current studies use side information from only one side, namely movie information.
Our StaTNA framework can be extended to incorporate side information for both movies and users: DISPLAYFORM0 where U and M denotes users and movies respectively.
Moreover, our StaTNA framework is also compatible with neural networks by including nuclear norm and sparse penalties to the objective.
We believe that similar formulations will provide us with better performance guarantees, but at the cost of model interpretability.
In this section, we discuss the methodologies we use to solve TNR and StaTNA.
As mentioned earlier, we use (Fast) Iterative Shrinkage-Thresholding Algorithm (FISTA, BID2 ) to solve these problems.
Before we address the detailed applications of these algorithms in our context to solve TNR and StaTNA, we introduce the following optimization oracles.
We define the proximal mapping of the 1 norm as DISPLAYFORM1 , whose extension to matrices is obtained by applying the scalar operator to each element.
Moreover, we define the proximal mapping of the nuclear norm BID4 BID13 DISPLAYFORM2 V , and Y = U DV is the SVD of matrix Y .
Now, using these definitions, we detail the algorithm to solve StaTNA in Algorithm 1.
Note that one can also initialize L 0 in both Algorithm 1 as DISPLAYFORM3 , where † denotes the pseudo-inverse of a matrix.For StaTNA, we directly apply FISTA to estimate L and S, and the procedures are detailed in Algorithm 1.
As aforementioned, TNR is a special case for StaTNA, so to solve TNR, we simply take λ S = ∞ in Algorithm 1, which forces all S k andŜ k to 0.
DISPLAYFORM4

assistant: text: Methodologies for recommender systems with side information based on trace-norm regularization