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Θ k × eoe h oe aaeeso neato function interaction of parameters model the denotes m eoeteltn etr arxo tm,where items, of matrix feature latent the denote i n item and R I P II. r ˆ i i.e. i,j i ioAntulov-Fantulin Nino i.e. steit o fmatrix of row i-th the is uih Switzerland Zurich, = RELIMINARIES v j i j [email protected] t o fmatrix of row -th f as T Zurich ETH ( R ∈ p i q , n j R | Let . Θ) R ∈ , p P i n j × eoe h latent the denotes R ∈ m P i.e. h original The . iial,let Similarly, . j j n R t column -th sthe is × k i.e. ervised raction denote zation ,our k, u we , j ons i ng. (1) -th ric ta- by ng ed as al ∈ s. y h - - - + + The latent features pi, qj and parameters Θ are found by representation qj is the j-th column of Q1 ≈ g(S2Q2 ). This minimizing the objective function model falls into the category of semi-NMF factorization mod- els [4]. Semi-NMF is a variant of NMF, which imposes non- L = L(rij , rˆij )+ λΩ(Θ), (2) negativity constraints only on the latent factors of the second i,j∈κ X layer Q+. This allows both positive and negative offsets from where L(.) is a point-wise (for pair-wise learning the bias term B. Usually, in clustering P represents cluster see [3]), Ω(.) is a regularizer (L2 or L1 norm [13]) and κ centroids and Q represents soft membership indicator [4]. But, denotes sets of training instances over which learning is done from recommender perspective, matrix P may be interpreted (see [22], about the missing data assumptions). as the coefficients and the nonnegativity The regularized matrix factorization [9], [12] approach is constraints imposed on latent item attributes Q allow part- modeling the interaction function f as the linear combination based representations [20], [21]. Note, that much of the observed variation in rating values is f(pi, qj |Θ) = k pi(k)qj (k), which represent the inner product in the latent space. The latent factors pi, qj are linear due to the effects associated with either users or items, known representations, dueP to the absence of non-linearity in the as biases or intercepts, independent of any interactions. Thus, linear algebra transformations R ≈ PQ. Here, usually L2 instead of explaining the full rating value by an interaction of T function is used for loss and regularization, while for the the form pi.qj , the system can try to identify the portion of training set κ, set of observed ratings were used. these values that individual user or item biases cannot explain. Neural collaborative filtering [6] models the interac- Bias bui involved in a rating rui can be computed as follows: tion function f as a deep neural network f(pi, qj |Θ) = bui = µ + bu + bi, where µ is the overall average rating, bu φout(φx(...(φ1(pi, qj ))...)), where φout and φx are mapping of is the observed deviations of user u and bi is the observed output layer and x-th layer of the form φ(z)= σ(Wz+b). The deviations of item i. σ(.) represents the non-linear function such as rectified linear Note, that in general case, we are able to compose + unit, hyperbolic tangent or others. The model parameters Θ more non-linearities with the following relation Qj ≈ + and latent factors pi, qj are learned in a joint manner, where g(Sj+1Qj+1), which generates the following model R ≈ + the latent factors are non-linear representations of user and B + P1g(S2g(S3g(...g(Sf Qf )))). However, more complex item vectors ui, vj from rating matrix R. item latent representation f ≥ 3 were showed not to be useful, Deep matrix factorizations [29] model the interaction func- see experiments section for more details. tion f as an inner product f(p , q |Θ) = p (k)q (k). i j k i j A. Learning Model Parameters However, both user and item features are deep representa- P tions of the form: pi = φout(φx(...(φ1(ui))...)) and qj = For a two layered item features structure, the model pa- φout(φx(...(φ1(vj ))...)). rameters are updated through an element-wise gradient de- scent approach, minimizing eq.(2), with squared loss function 2 III. PROPOSED MODEL L(rij , rˆij ) = (rui − rˆui) and L2 regularization Ω(Θ) = 2 2 2 2 2 In order to learn incomplete ratings, we formulate bu + bi + kpuk + kqik + ksuik . the following multilayer semi-NMF model (NSNMF): The model parameters Θ are randomly initialized uniformly ≈ + n×m at the range [0,1], and we perform iterative updates for each R B + P1Q1 , where R ∈ R is the rating matrix, n×m n×k B ∈ R is the bias matrix, P1 ∈ R is the latent observed rating rui as follows: user preferences matrix and Q+ ∈ Rk×m is the matrix 1 + ∗ of the non-linear item latent features. To model non-linear bu ← bu + η(eui − λbu) (4) representation of the items, we use the following model ∗ + + bi ← bi + η(eui − λbi) (5) Q1 ≈ g(S2Q2 ), which finally leads to: ∗ + puk ← puk + η eui · g skh · qhi − λpuk (6) R ≈ B + P1g(S2Q2 ), (3) h ! ! + l×m X where Q2 ∈ R+ is the hidden latent representation of items, ∗ ′ S2 is the weighting matrix between latent representations skl ← skl + η eui · puk · g skh · qhi qli − λskl , on different levels and g(.) is the element-wise non-linear h ! ! function to better approximate the non-linear manifolds of the X (7) ∗ ∗ latent features. (update only if g(S[k.]Q[.i]) > 0) Similar architecture to our proposed method without bias term is used in the task of learning deep representation of im- ∗ ′ q ← qli + η eui · puk · g skh · qhi skl − λqli , ages [27], where the learning depends on dense multiplicative li h ! ! updates. X (8) ∗ Simply, we model the interaction function f as an inner (update only if qli > 0), product with offset f(pi, qj |Θ) = bi,j + k pi(k)·qj (k), where where g’(.) is the derivative of activation function and eui = user feature pi is the i-th row of matrix P1 and item non-linear rui − rˆui is the error term. Note, that we do not explicitly P store the dense matrix B. The computational complexity for entropy loss function to predict scaled ratings on a continuous training a 2-layer item-feature NSNMF architecture is of order scale [0,1]. O(t(m + n)(kl + kl2)), where k,l are the dimensions of As for our proposed NSNMF method, we evaluate it with layer S2, and t the number of iterations. The different activation functions as follows: (i) NSNMF ReLU η was configured with the AdaGrad method [5], performing is the NSNMF model with Rectified Linear unit: ReLU(x)= larger updates for infrequent and smaller updates for frequent max(x, 0) as the activation function, (ii) NSNMF SoftPlus parameters. For this reason, it is well-suited for dealing with uses softplus: softplus(x) = log(1 + ex) as the activation sparse data, as in our case of incomplete ratings. Given k,l function and (iii) NSNMF ReLU bias is the proposed model are constant and (m+n) ≫ k,l, the scalability of the proposed with rectified linear unit activation function plus bias. method linearly depend on the number of users and items. In the supervised task, since we focus on explicit ratings, the rooted mean square error is used to assess the rate prediction IV. EXPERIMENTAL EVALUATION 1 2 0.5 performance [7]: RMSE = ( N (rui − rˆui) ) . The less In this section, we report experimental results by evaluating (u,i) the value of RMSE, the better theP approach performs. our NSNMF approach and a variety of baselines in both In the unsupervised task, we aim to inspect the difference supervised and unsupervised tasks. in representations obtained by our NSNMF and baseline A. Datasets approaches. We choose an unsupervised clustering task on such representations and thus use the pooled within-cluster We have three real datasets as follows: sum of squares around the cluster means (WCSS) [25]: • MovieLens100K 1 - 100004 ratings, across 9125 movies WCSS k 1 d , where n denotes the = r=1 2nr i,j∈Cr i,j r from 671 users. (have users with at least 20 ratings) number of elements inside of cluster Cr and di,j is the • FilmTrust 2 - 28496 ratings, across 1981 movies from euclidean distanceP betweenP instances i and j within the same 654 users. (filtered to have users with at least 20 ratings) cluster. • Amazon Music 3 - 50395 ratings, across 1188 items from 19260 users. (filtered to have users with at least 20 ratings C. Supervised tasks and items with at least 2 interactions) In the supervised task, we perform 10-fold cross-validation Each dataset is split into training (80%), and testing sets error for each dataset to determine the dimensions of (20%). The training set is then used for 10-fold cross- the hidden representation and regularizing parameter for validation for hyperparameter tuning. each approach. The dimensions of hidden representation were determined from cross-validation result for values in B. Baselines and setup {4,6,8,10,15,20}. The learning rate and regularizer value were We use baselines including both linear and nonlinear ap- varied in the range of {0.1, 0.01, 0.001}. proaches as follows: The final models were trained with learning rate 0.01 and CF Neighborhood models are the most common approach regularizing parameter 0.1 and factors value of 4,6,8 for to CF with user-oriented and item-oriented methods [15], [16]. FilmTrust, AMusic and MovieLens datasets respectively. They are respectively referred to as User-User CF and Item- Then, we report the rate prediction RMSE errors in Table Item CF. I. SVD is applied in the collaborative filtering domain by In Table I, we observe that NSNMF based approaches, i.e. factorizing the user-item rating matrix [17] by updating only NSNMF ReLU, Softplus and ReLU bias outperformed base- for the known ratings. lines across all datasets. Especially, NSNMF with ReLU bias performed the best and achieved up to less RMSE. NMF Nonnegative matrix factorization (NMF) which intro- 20% duces the nonnegative constraint into a MF process [12], [30] Meanwhile, DMF which learns non-linear representations has lower RMSE than the baselines based on linear transformation along with the regularized NMF mode to avoid over-fitting. i.e. User-User CF, Item-Item CF, SVD, NMF and regularized RBM Restricted Boltzmann Machine (RBM) [14] is an NMF in most of the time. We trained 4 DMF [29] using undirected , which contains a layer of visible normalized cross entropy loss on both implicit and explicit softmax units for items and a hidden binary unit for user ratings. The predicted ratings Rˆ on the scale [0,1], when scaled rating. Each hidden unit could then learn to model a significant to original scale [0,max(R)], where max(R) denotes the max dependency between the ratings of different movies. score in all ratings, perform worse than baselines compared to DMF presents a deep structure learning architecture to learn real ratings on the same scale with RMSE measure. Thus, we deep low dimensional representations respectively for users use their DMF architecture, trained using squared loss function and items [29]. They use both explicit ratings and implicit to predict unscaled ratings, which are then evaluated again feedback for a better optimization of a normalized cross with RMSE. 1https://grouplens.org/datasets/movielens/100k/, generated on October 17, Furthermore, we trained our model with different numbers 2016. of hidden layers to assess the prediction errors on all three 2https://www.librec.net/datasets.html 3http://jmcauley.ucsd.edu/data/amazon/ 4https://github.com/RuidongZ/Deep Matrix Factorizatio Models TABLE I TEST RMSE

NMF_3d NMF_8d Algorithm FilmTrust ML100K AMusic NSNMF_3d NSNMF_8d DMF_3d DMF_8d User-User CF 0.963 1.005 1.011 Item-Item CF 0.822 1.001 0.934 SVD 1.006 1.018 2.024 NMF 0.845 0.954 1.001 Regularized NMF 0.840 0.937 0.975 RBM 0.918 1.008 1.104 DMF 0.821 0.948 0.946

NSNMF ReLU 0.816 0.904 0.889 Within cluster sum of squares NSNMF Softplus 0.804 0.896 0.871 NSNMF ReLU bias 0.788 0.887 0.836 0 10000 20000 30000 40000 50000 60000 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 k (number of clusters) datasets. We found that 2-layer architecture better models Fig. 1. The pooled within-cluster sum of squares around the cluster means the variation in the rating matrix, while deeper layers even (WCSS) of clustering on MovieLens100k Dataset (where 3d, 8d denotes the decrease the performance. Due to the page limitation, we dimension of feature space) report the results up to 3 layers in Table II. algorithms [6], [28] that are training only on TABLE II implicit feedback. Thus, the current paper includes the deep TEST RMSE OF NSNMF W.R.T. DIFFERENTNUMBEROFLAYERS learning methods that use explicit feedback in their training algorithms [14], [29]. Algorithm FilmTrust ML100K AMusic ReLu 2 layer 0.816 0.904 0.889 We find out that simple linear regression over non-linear ReLu 3 layer 0.842 0.938 0.932 item representations is sufficient to overcome the performance of other deep learning methods that use explicit feedback in their training algorithms [14], [29]. It is important to stress, D. Unsupervised task that in our model the linear regression and non-linear item rep- In this part, we perform an unsupervised K-means clustering resentations are learned in a joint manner via non-linear semi method to evaluate the item representation learned by different non-negative matrix factorization. The non-negative constraint approaches in latent spaces. We performed each approach with allows better interpretability of item features e.g. movie cannot the hyperparameter set via cross-validation and then obtain the have a negative number of certain actors, a negative indication derived representation. to certain genre etc. However, the semi non-negativity con- In Figure 1, we report the WCSS [25] of each approach straint allows the regression coefficients to become negative w.r.t. the number of clusters. We observe that our NSNMF e.g. negative relation to certain item features. ReLu and DMF constantly yield lower WCSS than NMF. Furthermore, the linear interaction of non-linear item fea- It suggests that representation derived by non-linear matrix tures provides better predictions than the combination of non- factorization demonstrates higher representation ability. The liner item and non-linear user features, as in the case of Deep WCSS of NSNMF ReLu and DMF are quite comparable. It Matrix Factorization model [29]. indicates that non-linear transformation is the dominant part VI. CONCLUSIONS while the way of the combination of such representation results in a minor difference in the derived representation. Moreover, We introduced a multilayer nonlinear semi-nonnegative ma- the simple linear combination of non-linear representation trix factorization method to learn from the incomplete rating leads to better generalization ability in supervised prediction, matrix. The multilayer approach, which automatically learns which is already demonstrated in Table I. the hierarchy of attributes of the items, as well as the non- negative constraint help in the better interpretation of these V. DISCUSSION AND FUTURE WORK factors. Furthermore, we presented an algorithm for optimizing In this paper, we focus on learning the non-linear item the factors of our architecture with different non-linearities. representations for the explicit feedback and left the extension We evaluate our approach in comparison to a variety of of learning non-linear implicit feedback representations for matrix factorization and deep learning baselines using both future work. Most of the deep learning architectures have supervised rate prediction and unsupervised clustering in latent been implemented using dense implicit feedback rating matrix. item space. The results offer the insights as follows: (i) simple In this paper, we implement the proposed architecture for linear combination of non-linear representations realized in explicit feedback only, and left the implementation on implicit our proposed approach achieves better generalization ability, feedback for future work. We believe it will be interesting to that is, lower errors in the prediction on hold-out datasets. see the performance of the proposed algorithm on implicit (ii) in the unsupervised clustering task, we find out that the feedback, which will provide a better comparison with the representations learned by our approach yield comparable clustering performance metric (within-cluster sum of squares) [15] Sarwar, Badrul, et al. ”Item-based collaborative filtering recommenda- as deep matrix factorization. tion algorithms.” Proceedings of the 10th international conference on World Wide Web. ACM, 2001. Acknowledgement [16] Karypis, George. ”Evaluation of item-based top-n recommendation algo- N.A.-F. and T.G. are grateful for financial support from the rithms.” Proceedings of the tenth international conference on Information EU Horizon 2020 project SoBigData under grant agreement and knowledge management. ACM, 2001. [17] Sarwar, Badrul, et al. Application of in rec- No. 654024. 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