Abstract
The quality of online services highly depends on the accuracy of the recommendations they can provide to users. Researchers have proposed various similarity measures based on the assumption that similar people like or dislike similar items or people, in order to improve the accuracy of their services. Additionally, statistical models, such as the stochastic block models, have been used to understand network structures. In this paper, we discuss the relationship between similarity-based methods and statistical models using the Bernoulli mixture models and the expectation-maximization (EM) algorithm. The Bernoulli mixture model naturally leads to a completely positive matrix as the similarity matrix. We prove that most of the commonly used similarity measures yield completely positive matrices as the similarity matrix. Based on this relationship, we propose an algorithm to transform the similarity matrix to the Bernoulli mixture model. Such a correspondence provides a statistical interpretation to similarity-based methods. Using this algorithm, we conduct numerical experiments using synthetic data and real-world data provided from an online dating site, and report the efficiency of the recommendation system based on the Bernoulli mixture models.
Highlights
In this paper, we study recommendation problems, in particular, the reciprocal recommendation.The reciprocal recommendation is regarded as an edge prediction problem of random graphs.For example, a job recruiting service provides preferable matches between companies and job seekers.The corresponding graph is a bipartite graph, where nodes are categorized into two groups: job seekers and companies
Bernoulli mixture models (BMMs) have been employed in some studies [36,37,38] for block clustering problems, Here, we show that the BMMs are useful for recommendation problems
We considered the relationship between the similarity-based recommendation methods and statistical models
Summary
We study recommendation problems, in particular, the reciprocal recommendation. The recommendation system provides potentially preferable partners to each user The quality of such services depends entirely on the prediction accuracy of the unobserved or newly added edges. A similarity matrix defined from the similarity measure is used for the recommendation Another approach is employing the statistical models, such as stochastic block models [6], that are used to estimate network structures, such as clusters or edge distributions. Based on the above argument, we connect the similarity measures using completely positive matrices to the statistical models. We conduct numerical experiments using synthetic data and real-world data provided from an online dating site, and report the efficiency of the recommendation method based on the Bernoulli mixture models. Each directed graph, the adjacency matrix A node of the graph corresponds to each user with attributes such as the age, gender, preferences, etc. In real-world networks, the attributes are expected to be closely related to the graph structure
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