Normalized nonnegative models assign probability distributions to users and random variables to items; see [Stark, 2015]. Rating an item is regarded as sampling the random variable assigned to the item with respect to the distribution assigned to the user who rates the item. Models of that kind are highly expressive. For instance, using normalized nonnegative models we can understand users' preferences as mixtures of interpretable user stereotypes, and we can arrange properties of users and items in a hierarchical manner. These features would not be useful if the predictive power of normalized nonnegative models was poor. Thus, we analyze here the performance of normalized nonnegative models for top-N recommendation and observe that their performance matches the performance of methods like PureSVD which was introduced in [, 2010]. We conclude that normalized nonnegative models not only provide accurate recommendations but they also deliver (for free) representations that are interpretable. We deepen the discussion of normalized nonnegative models by providing further theoretical insights. In particular, we introduce total variational distance as an operational similarity measure, we discover scenarios where normalized nonnegative models yield unique representations of users and items, we prove that the inference of optimal normalized nonnegative models is NP-hard and finally, we discuss the relationship between normalized nonnegative models and nonnegative matrix factorization.
We mentioned related work when discussing similarity measures, time complexity and relation to NMF. Therefore, we describe here relations between NNMs and other models for item recommendation. We start with (PMF, see @cite_0 @cite_13 ) which forms intriguing family of models related to NNMs. In PMFs, the rating @math of user @math for item @math is regarded as Gaussian random variable. Mean and variance of @math are modeled in terms of @math and @math , respectively. Here, @math are low-dimensional vectors assigned to users and items. This structure is reminiscent of and . However, to apply PMF we need to assume that @math is Gaussian. In contrast, to apply NNMs, we do not need to assume anything about the distribution of @math and @math can be treated as random variable. The interpretability of PMFs is high because in principle, they allow for the computation of hierarchical orderings of properties of users and items (through the game introduced in @cite_14 ). But due to the infinite-dimensional nature of PMFs (i.e., @math ), the behavior of users cannot be interpreted easily through stereotypes; see @cite_14 .
@cite_0 @cite_14 @cite_13
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We introduce normalized nonnegative models (NNM) for explorative data analysis. NNMs are partial convexifications of models from probability theory. We demonstrate their value at the example of item recommendation. We show that NNM-based recommender systems satisfy three criteria that all recommender systems should ideally satisfy: high predictive power, computational tractability, and expressive representations of users and items. Expressive user and item representations are important in practice to succinctly summarize the pool of customers and the pool of items. In NNMs, user representations are expressive because each user's preference can be regarded as normalized mixture of preferences of stereotypical users. The interpretability of item and user representations allow us to arrange properties of items (e.g., genres of movies or topics of documents) or users (e.g., personality traits) hierarchically. Low-rank matrix approximation methods provide one of the simplest and most effective approaches to collaborative filtering. Such models are usually fitted to data by finding a MAP estimate of the model parameters, a procedure that can be performed efficiently even on very large datasets. However, unless the regularization parameters are tuned carefully, this approach is prone to overfitting because it finds a single point estimate of the parameters. In this paper we present a fully Bayesian treatment of the Probabilistic Matrix Factorization (PMF) model in which model capacity is controlled automatically by integrating over all model parameters and hyperparameters. We show that Bayesian PMF models can be efficiently trained using Markov chain Monte Carlo methods by applying them to the Netflix dataset, which consists of over 100 million movie ratings. The resulting models achieve significantly higher prediction accuracy than PMF models trained using MAP estimation.