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.
In section we have seen how normalized nonnegative models and nonnegative matrix factorization (NMF) are related to each other. NMF plays an important role in recommendation in general; see @cite_5 . Of course, probabilistic models can be regarded as a conveniently regularized NMF but that perspective disregards the operational interpretation of the columns of the nonnegative matrices @math , @math that constitute the NMF @math . This interpretation is important and the choice of regularization of an NMF affects the predictive performance of the studied recommender system.
@cite_5
2170293972 1969013340 2146913572 2144359569
Non-negative matrix factorization (NMF), i.e. V ap WH where both V, W and H are non-negative has become a widely used blind source separation technique due to its part based representation. The NMF decomposition is not in general unique and a part based representation not guaranteed. However, imposing sparseness both improves the uniqueness of the decomposition and favors part based representation. Sparseness in the form of attaining as many zero elements in the solution as possible is appealing from a conceptional point of view and corresponds to minimizing reconstruction error with an L0 norm constraint. In general, solving for a given L0 norm is an NP hard problem thus convex relaxation to regularization by the L1 norm is often considered, i.e., minimizing (1 2||V - WH||F 2 + lambda||H||1).An open problem is to control the degree of sparsity lambda imposed. We here demonstrate that a full regularization path for the L1 norm regularized least squares NMF for fixed W can be calculated at the cost of an ordinary least squares solution based on a modification of the least angle regression and selection (LARS) algorithm forming a non-negativity constrained LARS (NLARS). With the full regularization path, the L1 regularization strength lambda that best approximates a given L0 can be directly accessed and in effect used to control the sparsity of H. The MATLAB code for the NLARS algorithm is available for download. Although nonnegative matrix factorization (NMF) is NP-hard in general, it has been shown very recently that it is tractable under the assumption that the input nonnegative data matrix is close to being separable. (Separability requires that all columns of the input matrix belong to the cone spanned by a small subset of these columns.) Since then, several algorithms have been designed to handle this subclass of NMF problems. In particular, [Adv. Neural Inform. Process. Syst., 25 (2012), pp. 1223--1231] proposed a linear programming model, referred to as Hottopixx. In this paper, we provide a new and more general robustness analysis of their method. In particular, we design a provably more robust variant using a postprocessing strategy which allows us to deal with duplicates and near duplicates in the data set. We propose a novel nonnegative matrix factorization model that aims at finding localized, part-based, representations of nonnegative multivariate data items. Unlike the classical nonnegative matrix factorization (NMF) technique, this new model, denoted "nonsmooth nonnegative matrix factorization" (nsNMF), corresponds to the optimization of an unambiguous cost function designed to explicitly represent sparseness, in the form of nonsmoothness, which is controlled by a single parameter. In general, this method produces a set of basis and encoding vectors that are not only capable of representing the original data, but they also extract highly focalized patterns, which generally lend themselves to improved interpretability. The properties of this new method are illustrated with several data sets. Comparisons to previously published methods show that the new nsNMF method has some advantages in keeping faithfulness to the data in the achieving a high degree of sparseness for both the estimated basis and the encoding vectors and in better interpretability of the factors. Nonnegative Matrix Factorization (NMF), a relatively novel paradigm for dimensionality reduction, has been in the ascendant since its inception. It incorporates the nonnegativity constraint and thus obtains the parts-based representation as well as enhancing the interpretability of the issue correspondingly. This survey paper mainly focuses on the theoretical research into NMF over the last 5 years, where the principles, basic models, properties, and algorithms of NMF along with its various modifications, extensions, and generalizations are summarized systematically. The existing NMF algorithms are divided into four categories: Basic NMF (BNMF), Constrained NMF (CNMF), Structured NMF (SNMF), and Generalized NMF (GNMF), upon which the design principles, characteristics, problems, relationships, and evolution of these algorithms are presented and analyzed comprehensively. Some related work not on NMF that NMF should learn from or has connections with is involved too. Moreover, some open issues remained to be solved are discussed. Several relevant application areas of NMF are also briefly described. This survey aims to construct an integrated, state-of-the-art framework for NMF concept, from which the follow-up research may benefit.