A popular approach to collaborative filtering is matrix factorization. In this paper we develop a non-linear probabilistic matrix factorization using Gaussian process latent variable models. We use stochastic gradient descent (SGD) to optimize the model. SGD allows us to apply Gaussian processes to data sets with millions of observations without approximate methods. We apply our approach to benchmark movie recommender data sets. The results show better than previous state-of-theart performance. 1.

Rating prediction is an important application, and a popular research topic in collaborative filtering. However, both the validity of learning algorithms, and the validity of standard testing procedures rest on the assumption that missing ratings are missing at random (MAR). In this paper we present the results of a user study in which we collect a random sample of ratings from current users of an online radio service. An analysis of the rating data collected in the study shows that the sample of random ratings has markedly different properties than ratings of user-selected songs. When asked to report on their own rating behaviour, a large number of users indicate they believe their opinion of a song does affect whether they choose to rate that song, a violation of the MAR condition. Finally, we present experimental results showing that incorporating an explicit model of the missing data mechanism can lead to significant improvements in prediction performance on the random sample of ratings. 1

parkst @ yahoo-inc.com We propose a new ranking method, which combines recommender systems with information search tools for better search and browsing. Our method uses a collaborative filtering algorithm to generate personal item authorities for each user and combines them with item proximities for better ranking. To demonstrate our approach, we build a prototype movie search and browsing engine called MAD6 (Movies, Actors and Directors; 6 degrees of separation). We conduct offline and online tests of our ranking algorithm. For offline testing, we use Yahoo! Search queries that resulted in a click on a Yahoo! Movies or Internet Movie Database (IMDB) movie URL. Our online test involved 44 Yahoo! employees providing subjective assessments of results quality. In both tests, our ranking methods show significantly better recall and quality than IMDB search and Yahoo! Movies current search.

With the sheer growth of online user data, it becomes challenging to develop preference learning algorithms that are sufficiently flexible in modeling but also affordable in computation. In this paper we develop nonparametric matrix factorization methods by allowing the latent factors of two low-rank matrix factorization methods, the singular value decomposition (SVD) and probabilistic principal component analysis (pPCA), to be data-driven, with the dimensionality increasing with data size. We show that the formulations of the two nonparametric models are very similar, and their optimizations share similar procedures. Compared to traditional parametric low-rank methods, nonparametric models are appealing for their flexibility in modeling complex data dependencies. However, this modeling advantage comes at a computational price — it is highly challenging to scale them to large-scale problems, hampering their application to applications such as collaborative filtering. In this paper we introduce novel optimization algorithms, which are simple to implement, which allow learning both nonparametric matrix factorization models to be highly efficient on large-scale problems. Our experiments on EachMovie and Netflix, the two largest public benchmarks to date, demonstrate that the nonparametric models make more accurate predictions of user ratings, and are computationally comparable or sometimes even faster in training, in comparison with previous state-of-the-art parametric matrix factorization models.

A fundamental aspect of rating-based recommender systems is the observation process, the process by which users choose the items they rate. Nearly all research on collaborative filtering and recommender systems is founded on the assumption that missing ratings are missing at random. The statistical theory of missing data shows that incorrect assumptions about missing data can lead to biased parameter estimation and prediction. In a recent study, we demonstrated strong evidence for violations of the missing at random condition in a real recommender system. In this paper we present the first study of the effect of non-random missing data on collaborative ranking, and extend our previous results regarding the impact of non-random missing data on collaborative prediction.

Discrete mixed membership modeling and continuous latent factor modeling (also known as matrix factorization) are two popular, complementary approaches to dyadic data analysis. In this work, we develop a fully Bayesian framework for integrating the two approaches into unified Mixed Membership Matrix Factorization (M 3 F) models. We introduce two M 3 F models, derive Gibbs sampling inference procedures, and validate our methods on the EachMovie, MovieLens, and Netflix Prize collaborative filtering datasets. We find that, even when fitting fewer parameters, the M 3 F models outperform state-ofthe-art latent factor approaches on all benchmarks, yielding the greatest gains in accuracy on sparsely-rated, high-variance items. 1.

A general framework based on Gaussian models and a MAP-EM algorithm is introduced in this paper for solving matrix/table completion problems. The numerical experiments with the standard and challenging movie ratings data show that the proposed approach, based on probably one of the simplest probabilistic models, leads to the results in the same ballpark as the state-of-the-art, at a lower computational cost. Index Terms — Matrix completion, inverse problems, collaborative filtering, Gaussian mixture models, MAP estimation, EM algorithm

Optimization problems with a nuclear norm regularization, such as e.g. low norm matrix factorizations, have seen many applications recently. We propose a new approximation algorithm building upon the recent sparse approximate SDP solver of (Hazan, 2008). The experimental efficiency of our method is demonstrated on large matrix completion problems such as the Netflix dataset. The algorithm comes with strong convergence guarantees, and can be interpreted as a first theoretically justified variant of Simon-Funk-type SVD heuristics. The method is free of tuning parameters, and very easy to parallelize. 1.

Collaborative filtering (CF) is a data analysis task appearing in many challenging applications, in particular data mining in Internet and e-commerce. CF can often be formulated as identifying patterns in a large and mostly empty rating matrix. In this paper, we focus on predicting unobserved ratings. This task is often a part of a recommendation procedure. We propose a new CF approach called interlaced generalized linear models (GLM); it is based on a factorization of the rating matrix and uses probabilistic modeling to represent uncertainty in the ratings. The advantage of this approach is that different configurations, encoding different intuitions about the rating process can easily be tested while keeping the same learning procedure. The GLM formulation is the keystone to derive an efficient learning procedure, applicable to large datasets. We illustrate the technique on three public domain datasets. r 2008 Elsevier B.V. All rights reserved.

We present a Matrix Factorization (MF) based approach for the Netflix Prize competition. Currently MF based algorithms are popular and have proved successful for collaborative filtering tasks. For the Netflix Prize competition, we adopt three different types of MF algorithms: regularized MF, maximum margin MF and non-negative MF. Furthermore, for each MF algorithm, instead of selecting the optimal parameters, we combine the results obtained with several parameters. With this method, we achieve a performance that is more than 6 % better than the Netflix’s own system.