Consistent Collaborative Filtering via Tensor Decomposition

Understanding preferences based on users’ historical activities is key for personalized recommendation. This is particularly challenging when explicit ratings on many items are not available. In this scenario, historical activities are typically viewed as binary, representing whether a user has interacted with an item or not. Users’ preferences must be inferred from such implicit feedback with additional assumptions based on this binary data.

One common assumption is to view non-interacted items as negatives, meaning users are not interested in them; items that have been interacted are often assumed to be preferred ones (Hu et al., 2008; Pan et al., 2008). In reality however, such an assumption is rarely met. For example, lack of interaction between a user and an item might simply be the result of lack of exposure. It is therefore more natural to assume that non-interacted items are a combination of the ones that users do not like and the ones that users are not aware of (Rendle et al., 2009).

With this assumption, Rendle et al. (2009) proposed to give partial orders between items. Particularly, they assumed that items which users have interacted with are more preferable than the non-interacted ones. With this assumption in mind, Bayesian Personalized Ranking (BPR) was developed to perform personalized recommendations. In BPR, the observed data are transformed into a three-way binary tensor $D$ with the first dimension representing users. The other two dimensions represent items, encoding personalized preferences between item pairs (Figure 1(a)). Mathematically, this means that any first order slice of $D$ at the $u$-th user, $D_{u::}$, is represented as a pairwise comparison matrix (PCM) between items. The $(i,j)$-th entry when observed, $d_{uij}\in\{-1,1\}$, is binary, suggesting whether $u$-th user prefers ($d_{uij}=1$) the $i$-th item over the $j$-th one, or the other way around ($d_{uij}=-1$). The tensor $D$ is only partially observed, with missing entries where there is no prior knowledge to infer any preference for a particular user. The recommendation problem becomes finding a parsimonious parameterization of the generative model for observed entries in $D$ and estimating model parameters which best explain the observed data.

The model used in BPR assumes that among the observed entries in $D$, the probability that the $u$-th user prefers the $i$-th item over the $j$-th item can be modeled as a Bernoulli distribution (Hu et al., 2008),

where $X=\{x_{uij}\}\in\mathbb{R}^{n\times m\times m}$ is the collection of unknown parameters of the Bernoulli distribution (Figure 1(b)). In fact, $x_{uij}$ is the natural parameter of a Bernoulli distribution. Rendle et al. (2009) decompose the natural parameter by

In the equation, $x_{ui}$ ($x_{uj}$) can be interpreted as the strength of preference on the $i$-th ($j$-th) item for the $u$-th user. In other words, users’ strengths of preference on different items are represented by scalar values denoted as $x_{ui}$. The relative preferences between items are therefore characterized by the differences between their preference’s strengths for a particular user. With this decomposition, $x_{uij}$ becomes $u$-th user’s relative preference on the $i$-th item over the $j$-th item.

By further letting $x_{ui}$ be represented as a dot product between a $k$ dimensional user vector $\xi_{u}\in\mathbb{R}^{k}$ and an item vector $\eta_{i}\in\mathbb{R}^{k}$,

Rendle et al. (2009) reveal the connection between their proposed BPR model and traditional collaborative filtering models such as matrix factorization.

The factorization in equations 2 and 3 provides a parsimonious representation of the original Equation 1. Without the factorization, one could model $x_{uij}$ independently. This model requires $n\times m\times m$ free parameters, and a model fitting would result in $x_{uij}=\infty$ for $d_{uij}=1$, and $x_{uij}=-\infty$ for $d_{uij}=-1$. The rests of $x_{uij}$ with missing $d_{uij}$ are left unknown. The parsimonious representation, in contrast, reduces the number of parameters to $(n+m)\times k$. The value of $x_{uij}$ now becomes coupled with $x_{uit}$ for $t\neq j$, and entries in $D$ jointly impact parameter estimate of $x_{uij}$. This highlights a fundamental assumption behind collaborative filtering: information can be learned from items that share similar pattern when they are interacted by users, and one user can learn from another user who interacts with similar set of items.

Equation 3 has a direct link to the Bradley-Terry model often studied when analyzing a PCM for decision making (Hunter, 2004; Weng & Lin, 2011). This model can be at least dated back to 1929 (Zermelo, 1929). One property of the model is its transitivity: The relative preference $x_{uij}$ can be expressed as the sum of relative preferences of $x_{uit}$ and $x_{utj}$, for any user with $t\neq i$ and $t\neq j$. However, in reality this transitivity property is less frequently met. Only around $~{}3\%$ of real world PCM’s satisfy complete transitivity (Mazurek & Perzina, 2017). The violation of the property is more conceivable when users are exposed to various types of items. For example, in an online streaming platform, one favorable movie could become less intriguing after a subscriber watches a different style/genre.

In this paper, we extend the original BPR model (which is one of the most fundamental models in recommendation systems) to allow non-transitive user ratings. In particular, we extend Equation 2 to a more general form by proposing a new tensor decomposition. We denote our new model SAD (Sliced Anti-symmetric Decomposition). The new tensor decomposition introduces a second set of non-negative item vectors $\tau_{i}$ for every item. Different from the first vector $\eta_{i}$, the new vector contributes negatively when calculating relative preferences, producing counter-effects to the original strength of users’ preferences; see Section 4 for more details. Mathematically, the new vector extends the original dot product in Equation 3 to an inner product. The original BPR model becomes a special case of SAD when the values in $\tau_{i}$ are all set to $1$, and the inner product reduces to a standard dot product. When $\tau_{i}$ contains entries that are not $1$, the transitivity property no longer necessarily holds. While assigning an $l_{1}$ regularization to the entries in $\tau_{i}$ to encourage its values being $1$ to reflect prior beliefs, SAD is able to infer its unknown value from real world data. We derive an efficient group coordinate descent algorithm for parameter estimation of SAD. Our algorithm results in a simple stochastic gradient descent (SGD) producing fast and accurate parameter estimations. Through a simulation study we first demonstrate the expressiveness of SAD and efficiency of the SGD algorithm. We then compare SAD to seven alternative SOTA recommendation models in three publicly available datasets with distinct characteristics. Results confirm that our new model permits to exploit information and relations between items not previously considered, and provides more consistent and accurate results as we will demonstrate in this paper.

Inferring priority via pairwise comparison. The Bradley-Terry model (Hunter, 2004; Weng & Lin, 2011; Zermelo, 1929) has been heavily used along this line of research. In the Bradley-Terry model, the probability of the $i$-th unit (an individual, a team, or an item) being more preferable than the $j$-th unit (denoted as $i\succ j$) is modeled by

where $\lambda_{i}$ represents the strength, or degree, of preference of the $i$-th unit. The goal is to estimate $\lambda_{i}$ for all units based on pairwise comparisons. The link to Equation 1 becomes clear once we omitting user index and set $x_{i}=\log(\lambda_{i})$. In fact, the original BPR model can be viewed as an extension to the Bradley-Terry model to allow personalized parameters, and the strength of preferences are assumed to be dot products of user and item vectors as in Equation 3.

Various algorithms have been developed for parameter estimation of this model. For example, Hunter (2004) developed a class of algorithms named minorization-maximization (MM) for parameter estimation. In MM, a minorizing function $Q$ is maximized to find the next parameter update at every iteration. Refer to the work by Hunter & Lange (2004) for more details related to MM. Weng & Lin (2011) proposed a Bayesian approximation method to estimate team’s priorities from outputs of games between teams. Most recently, Wang et al. (2021) developed a bipartite graph iterative method to infer priorities from large and sparse pairwise comparison matrices. They applied the algorithm to the Movie-Lens dataset to rank movies based on their ratings aggregated from multiple users. Our paper is different from aforementioned models in that we model user-specific item preferences under personalized settings.

Tensor decompositions for recommendation. Compared to traditional collaborative filtering methods using matrix factorizations, tensor decompositions have received less attention in this field until recently. The BPR model can be viewed as one of the first attempts to approach the recommendation problems using tensor analysis. As discussed in Section 1, by making the assumption that interacted items are more preferrable compared to non-interacted ones, user-item implicit feedback are represented as a three-way binary tensor (Rendle et al., 2009). In their later work, the authors developed tensor decomposition models for personalized tag recommendation (Rendle & Schmidt-Thieme, 2010). The relationship between their approach and traditional tensor decomposition approaches such as Tucker and PARAFAC (parallel factors) decompositions (Kiers, 2000; Tucker, 1966) was discussed.

Recently, tensor decomposition methods have been used to build recommendation systems using information from multiple sources. Wermser et al. (2011) developed a context aware tensor decomposition approach by using information from multiple sources, including time, location, and sequential information. Hidasi & Tikk (2012) considered implicit feedback and incorporated contextual information using tensor decomposition. They developed an algorithm which scaled linearly with the number of non-zero entries in a tensor. A comprehensive review about applications of tensor methods in recommendation can be found by Frolov & Oseledets (2017) and references therein. Different from leveraging multiple sources of information, the SAD model developed in this paper considers the basic scenario where only implicit feedback are available, the scenario that is considered in BPR model (Rendle et al., 2009). Our novelty lies in the fact that we propose a more general form of tensor decomposition for modeling implicit feedback.

Deep learning in recommendation models. Deep learning has attracted significant attention in recent years, and the recommendation domain is no exception. Traditional approaches such as collaborative filtering and factorization machines (FM) have been extended to incorporate neural network components (Chen et al., 2017; He et al., 2017; Xiao et al., 2017). In particular, Chen et al. (2017) replaced the dot product that has been widely used in traditional collaborative filtering with a neural network containing Multilayer Perceptrons (MLP) and embedding layers. Chen et al. (2017) and Xiao et al. (2017) introduced attention mechanisms (Vaswani et al., 2017) to both collaborative filtering and FM (Rendle, 2010). Mostly recently Rendle et al. (2020) revisited the comparison of traditional matrix factorization and neural collaborative filtering and concluded that matrix factorization models can be as powerful as their neural counterparts with proper hyperparameters selected. Despite the controversy, various types of deep learning models including convolutional networks, recurrent networks, variational auto-encoders (VAEs), attention models, and combinations thereof have been successfully applied in recommendation systems. The work by Zhang et al. (2019) provided an excellent review on this topic. This line of research doesn’t have direct link to the SAD model considered in the current work. However, we provide a brief review along this line since our model could be further extended to use the latest advances in the area.

We use $n$ to denote the total number of users in a dataset and $m$ to denote the total number of items. Users are indexed by $u\in[1,\cdots n]$. Items are indexed by both $i$ and $j\in[1,\cdots m]$. We use $k$ to denote the number of latent factors, and use $h\in[1,\cdots,k]$ to index a factor. Capital letters are used to denote a matrix or a tensor, and lowercase letters to denote a scalar or a vector. For example, the three-way tensor of observations is denoted as $D\in\mathbb{R}^{n\times m\times m}$ and the $(u,i,j)$-th entry is denoted as $d_{uij}$. Similarly, the user latent matrix is denoted as $\Xi\in\mathbb{R}^{k\times n}$, and its $u$-th column is denoted as $\xi_{u}$ to represent the user vector for $u$-th user. We use $\xi^{h}$ to denote the $h$-th row (the $h$-th factor) of $\Xi$ viewed as a column vector.

We start with the original BPR model. The relative preference $x_{uij}$ defined in Equation 2 forms a three-way tensor $X\in\mathbb{R}^{n\times m\times m}$. The BPR model (Rendle et al., 2009) can be viewed as one parsimonious representation of tensor $X$. Let $\xi_{u}\in\mathbb{R}^{k}$ and $\eta_{i}\in\mathbb{R}^{k}$ denote the user and item vectors respectively, and let $\xi_{hu}$ ($\eta_{hi}$) indicate the $h$-th entry in $\xi_{u}$ ($\eta_{i}$). Equations 1, 2, and 3 can be re-written as

where $X_{u::}$ is the first order slice of $X$ at $u$-th user,

$\eta^{h}\in\mathbb{R}^{m}$ being the $h$-th row of item matrix $H\in\mathbb{R}^{k\times m}$. $\mathbb{1}\in\mathbb{R}^{m}$ is used to denote a vector of all $1$’s and $\circ$ being the outer product.

We first evaluate the performance of SAD and the SGD algorithm on simulation data, with the goal of examining the performance of our algorithm with true parameters known ahead. We choose $n=20$ users, $m=50$ items, and $k=5$, resulting in $\Xi\in\mathbb{R}^{5\times 20}$, $H\in\mathbb{R}^{5\times 50}$, and $T\in\mathbb{R}^{5\times 50}_{+}$. We consider two scenarios in the simulation. In the first simulation (Sim1) we set $T$ to $1$, effectively reducing SAD to the generative model of BPR (Rendle et al., 2009). In the second scenario (Sim2), we set a small proportion of $T$ to either $0.01$ or $5$, the other entries are set to $1$. For user matrix $\Xi$ and left item matrix $H$, their values are uniformly drawn from the interval $[-2,2]$. We calculate the preference tensor $X$ with Equation 8 and draw an observation tensor $D$ from the corresponding Bernoulli distributions.

We first examine the performance of SAD with complete observations in Sim2 to validate if our method is able to generate accurate parameter estimation. We run the SGD algorithm with a learning rate $0.05$. The weight of the $l_{1}$ regularization assigned to $T$ is set to $0.01$, and the weight of the $l_{2}$ regularization is set to $0.005$. Initial values of parameters are randomly drawn from a standard Gaussian distribution. The number of latent factors $k$ is set to the true value. In reality when $k$ is unknown, cross validation can be used to select the best value of $k$. After 20 epochs, $\hat{T}$ is able to recover the sparse structure of the true parameters of $T$ up to permutation of factors (Figure 2). The user matrix and left item matrix converge to the true parameter values as well (Figure 3).

Next we examine the performance of SAD in both Sim1 and Sim2, under the scenarios with missing data. To be more specific, we randomly mark $x\%$ of $D$ as missing to mimic missing at random. Note that in the real world, observations could have more complex missingness structures. As a comparison, we run BPR under the same contexts.

The convergences of SAD in Sim1 are shown in Figure 4(a), together with the estimated sparsity of $T$. Here the sparsity of $T$ is defined as the percentage of the entries in $T$ with $|\tau_{hj}-1|<0.05$. When the percentage of missingness is at small or medium levels, SAD is able to converge to a sparsity close to $1$, suggesting the effectiveness of the $l_{1}$ regularization. It becomes more challenging when the percentage of missingness surpasses $70\%$. Figure 4(c) shows the trajectories of the mean squared error (MSE) between $\hat{X}$ and the true $X$ under different missingness percentages for both SAD and BPR. Both SAD and BPR are able to converge to a low MSE with small/medium percentages of missingness. Similarly, with a high percentage of missingness, the performance of both models begins to deteriorate. We conclude that SAD has a performance on par with BPR when data are simulated from the generative model of BPR.

Results for Sim2 are shown in Figure 4(b). Note that the true sparsity of $T$ is $86\%$. SAD is able to generate an accurate estimation of the sparsity under small/medium percentages of missingness. When evaluating both models using MSE, SAD is able to achieve a much lower value due to its correct specification of the generative model (Figure 4(d)).

We select three real world datasets to evaluate SAD and compare against SOTA recommendation models. The datasets selected contain explicit integer valued ratings. Nonetheless, we mask their values and view them as binary. The explicit ratings are used as a means to evaluate models’ consistency in pairwise comparison after model fitting (details below). The first dataset used is from the Netflix Prize (Bennett et al., 2007). The original dataset contains movie ratings of $8,921$ movies from $478,533$ unique users, with a total number of ratings reaching to over $50M$. We randomly select $10,000$ users as our first dataset. The resulting dataset contains $8,693$ movies with over $1M$ ratings from the $10,000$ users. For the second dataset, we choose the Movie-Lens 1M dataset (Harper & Konstan, 2015). It contains over $1M$ ratings from $6,040$ users on $3,706$ movies. As a third dataset, we consider the reviews of recipes from Food-Com (Majumder et al., 2019). The complete dataset has $1.1M$ reviews from $227K$ users on $231K$ recipes. We select the top $20K$ users with the most activities, and filter out recipes receiving less than $50$ reviews. The resulting dataset has $145K$ reviews from $17K$ users (users with zero activity are further removed after filtering recipes) on $1.4K$ recipes.

The three datasets have distinct characteristics. Among the three datasets, the Food-Com review dataset has the least user-item interactions, even when the most active users/popular recipes are selected. The maximum number of items viewed by a single user is $878$. It also has the largest number of users. The Netflix dataset is the most skewed, with the number of items interacted by a user ranging from as low as $1$ to as high as over $8K$. The Movie-Lens dataset contains the largest number of user-item interactions, and is most uniformly distributed. Some details of the three datasets can be found in Table A in Appendix.

We choose seven SOTA recommendation models to compare with SAD. Their details are listed in Table C. For each of the model considered, we perform a grid search to determine hyperparameters. Models are chosen based on their goodness-of-fit using log likelihood. We evaluate the models using a comprehensive leave one interaction out (LOO) evaluation (Bayer et al., 2017; He et al., 2017; 2016), in which we randomly hold out one user-item interaction from training set for every user. Users who have only one interaction are skipped. We create $20$ such LOO sets for each dataset considered. The dimension of latent space during evaluation is set to $500$ for all models and datasets. The choice of the latent dimension can be further optimized using methods such as across validation. In this work we choose the same number across comparing SOTA models such that they can be evaluated on a common ground.

We consider two aspects of model’s performance during evaluation: consistency and recommendation. The consistency is defined as whether model’s prediction matches user’s actual pairwise preference. During evaluation, the hold out item $o$ and other items $j$ in interacted set $I_{u}$ of user $u$ are arranged such that $o\succ_{u}j$ based on users’ actual ratings. The mean of their predicted preference (Equation 11), the percentage of consistent predictions (Equation 12), and the median of per user percentage of consistency are reported in Table 6. SAD has the most consistent results among all eight recommendation models except in one scenario, in which our model is second best.