1. Graphbased models
Defining as the dataset of a Recommender System composed by the set of users , the set of items , the set of item features and all the relations among them. Defining as the set of vertices, as the set of edges and assigning to every edge a weight equal to the value of the relation between the nodes it connects, we can represent with a weighted tripartite graph . Recommendations on can be provided exploiting random walks over , accomplished by starting on a vertex and choosing randomly one of its neighbors at each step. We can represent with a square adjacency matrix , where every entry represents the weight of the edge that connects the node to . Normalizing rowwise, we can compute the probability, or markovian, matrix P where represents the probability of choosing any node when standing in any node . To get the probability relative to random walks of length , we can elevate to the power of . Finally, recommendation lists are provided selecting, for each users, the items with highest probabilities. It has been shown in literature that short random walks achieve best performance in most situations(Cooper et al., 2014). In particular, if we consider paths of length 3 over , we can distinguish two types, starting from a user node:

useritemuseritem: a collaborative path that exploits only interactions to reach the destination. Its results are obtained removing edges from items to features before elevating P. This approach has been called (Cooper et al., 2014).

useritemfeatureitem: a contentbased path that exploits item features over other users’ interactions to reach the destination. Its results are obtained removing edges from items to users before elevating .
2. Model simplification
In previous sections we stated that a graphbased random walk model uses only final useritem probabilities to produce recommendation lists, which means that of the whole 3 steps random walk result we only need the one referred to complete paths from users to items, that is represented by one single submatrix of . This property allows to reduce the exponentiation of to a multiplication of three of its submatrices. We define four nonzero submatrices of that can be derived directly from the feedback matrix and the binary item content matrix :

probability to reach an item node from a user node

: probability to reach a user node from an item node

: probability to reach a feature node from an item node

: probability to reach an item node from a feature node
The estimated useritem probabilities used for recommendations, that we will call , can be obtained with two different multiplications of these submatrices, depending on the nature of the path:

Collaborative:

Contentbased:
3. Feature weighting model
We can introduce feature weights over edges that connect items to features, obtaining a variant of that we will call . This way we can influence the probability to reach feature nodes and, consequently, other item nodes in the last steps of contentbased paths. Note that weights have to be strictly positive, because they directly become probabilities. As we stated in previous sections, we want to exploit collaborative information to estimate feature weights, so, in order to do that, we want to obtain as similar results as possible between the collaborative path and the weighted contentbased one. In other words we want that:
Now we can define a regression problem over feature weights to solve the equation, minimizing the residual sum of squares with Stochastic Gradient Descent. Given two items
, we can formalize the problem as:4. Target matrix
We can state that our model is the solution to a regression problem where the target is a probability matrix obtained with 3step random walks following the collaborative path. We will refer to this solution as the hybrid path. However, we could use as target any probability matrix that contains collaborative information. In particular, we will see the results obtained using the probability matrix of the (Paudel et al., 2017) approach, calculated adapting the popularitybased reranking procedure. We will refer to this alternative as the reranked hybrid path.
5. Datasets
We tested our model on the well known Movielens 20M dataset, using genres and lemmatized 1 and 2grams of user based tags as features, and on The Movies Dataset, publicly available on Kaggle^{1}^{1}1https://www.kaggle.com/rounakbanik/themoviesdataset, that adds to the Full Movielens dataset the editorial items metadata available on TMDb. To remove some noise, we applied some filters on both the datasets: we removed items and users with too few interactions, items with too few or too many features, and too rare features. We split each dataset for train and test, in order to keep a test set that could reproduce a cold start scenario. So we kept the 20% of the items, chosen randomly, and all their ratings for the test set, while we used the remaining 80% for the train phase. Then we split the training set in two more sets with the same 8020 proportions, respectively for the training and the validation of the model.
6. Evaluation
For the evaluation we used three common metrics of Recommender Systems literature, that can highlight the accuracy of the model in prediction and ranking: we used the @5 variants of Recall, Mean Average Precision and Normalized Discounted Cumulative Gain. We compared the results obtained by different approaches:

CBF: Contentbased KNN algorithm with cosine similarity

CBFIDF: Contentbased KNN algorithm with cosine similarity, using IDF to assign feature weights

: 3steps random walks following contentbased path

: 3steps random walks following hybrid path

: 3steps random walks following hybrid reranked path
7. Results






CBF  0.10135  0.22026  0.13957  
CBFIDF  0.10147  0.21813  0.13941  
0.08853  0.20617  0.12359  
0.09561  0.21323  0.13247  
0.10919  0.22826  0.14782 






CBF  0.04850  0.08727  0.06594  
CBFIDF  0.05033  0.08988  0.06720  
0.05482  0.09987  0.07298  
0.06810  0.12176  0.09118  
0.06776  0.11934  0.09017 
Analyzing results summarized in Tables 2 and 2, we can see that both the hybrid paths outperform the purely contentbased one, which means that the collaborative information exploited is useful and increases performance. We can also notice that the target matrix influences the quality of the final model. In particular, the reranked path provides more reliable results, and its performance is higher than both Contentbased KNN approaches on both datasets. The non reranked path, instead, is not able to reach CBF scores on Movielens 20M, but obtains the best results on The Movies Dataset. In conclusion, we can state that the model was able to outperform both nonweighted and IDF feature weighting approaches, showing the importance of collaborative information and proving to be a potentially good solution for coldstart scenarios.
8. Conclusion
We proposed a new approach to face the item coldstart problem of Recommender Systems. We have shown that it is possible to model a hybrid graphbased recommender exploiting collaborative information to estimate feature weights and improve quality of contentbased recommendations. Future directions include validating this results on more datasets and baselines, as well as learning from other collaborative probability matrices.
References
 (1)
 Cella et al. (2017) Leonardo Cella, Stefano Cereda, Massimo Quadrana, and Paolo Cremonesi. 2017. Derive item features relevance from past user interactions. In UMAP.
 Cooper et al. (2014) Colin Cooper, Sang Hyuk Lee, Tomasz Radzik, and Yiannis Siantos. 2014. Random walks in recommender systems: exact computation and simulations. In Proceedings of the 23rd International Conference on World Wide Web. ACM, 811–816.
 Paudel et al. (2017) Bibek Paudel, Fabian Christoffel, Chris Newell, and Abraham Bernstein. 2017. Updatable, accurate, diverse, and scalable recommendations for interactive applications. ACM Transactions on Interactive Intelligent Systems (TiiS) (2017).
 Sharma et al. (2015) Mohit Sharma, Jiayu Zhou, Junling Hu, and George Karypis. 2015. Featurebased factorized Bilinear Similarity Model for ColdStart Topn Item Recommendation. In Proceedings of the 2015 SIAM International Conference on Data Mining, Vancouver, BC, Canada, April 30  May 2, 2015. 190–198.
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