Matrix factorization is a fundamental technique in machine learning that is applicable to collaborative filtering, information retrieval and many other areas. In collaborative filtering and many other tasks, the objective is to fill in missing elements of a sparse data matrix. One of the biggest challenges in this case is filling in a column or row of the matrix with very few observations. In this paper we introduce a Bayesian matrix factorization model that performs regression against side information known about the data in addition to the observations. The side information helps by adding observed entries to the factored matrices. We also introduce a nonparametric mixture model for the prior of the rows and columns of the factored matrices that gives a different regularization for each latent class. Besides providing a richer prior, the posterior distribution of mixture assignments reveals the latent classes. Using Gibbs sampling for inference, we apply our model to the Netflix Prize problem of predicting movie ratings given an incomplete user-movie ratings matrix. Incorporating rating information with gathered metadata information, our Bayesian approach outperforms other matrix factorization techniques even when using fewer dimensions.

As more images and categories become available, organizing them becomes crucial. We present a novel statistical method for organizing a collection of images into a tree-shaped hierarchy. The method employs a non-parametric Bayesian model and is completely unsupervised. Each image is associated with a path through a tree. Similar images share initial segments of their paths and therefore have a smaller distance from each other. Each internal node in the hierarchy represents information that is common to images whose paths pass through that node, thus providing a compact image representation. Our experiments show that a disorganized collection of images will be organized into an intuitive taxonomy. Furthermore, we find that the taxonomy allows good image categorization and, in this respect, is superior to the popular LDA model.

In this paper we introduce a novel collapsed Gibbs sampling method for the widely used latent Dirichlet allocation (LDA) model. Our new method results in significant speedups on real world text corpora. Conventional Gibbs sampling schemes for LDA require O(K) operations per sample where K is the number of topics in the model. Our proposed method draws equivalent samples but requires on average significantly less then K operations per sample. On real-word corpora FastLDA can be as much as 8 times faster than the standard collapsed Gibbs sampler for LDA. No approximations are necessary, and we show that our fast sampling scheme produces exactly the same results as the standard (but slower) sampling scheme. Experiments on four real world data sets demonstrate speedups for a wide range of collection sizes. For the PubMed collection of over 8 million documents with a required computation time of 6 CPU months for LDA, our speedup of 5.7 can save 5 CPU months of computation.

A general modeling framework is proposed that unifies nonparametric-Bayesian models, topic-models and Bayesian networks. This class of infinite state Bayes nets (ISBN) can be viewed as directed networks of ‘hierarchical Dirichlet processes’ (HDPs) where the domain of the variables can be structured (e.g. words in documents or features in images). We show that collapsed Gibbs sampling can be done efficiently in these models by leveraging the structure of the Bayes net and using the forward-filtering-backward-sampling algorithm for junction trees. Existing models, such as nested-DP, Pachinko allocation, mixed membership sto- chastic block models as well as a number of new models are described as ISBNs. Two experiments have been performed to illustrate these ideas.