Hierarchical Compound Poisson Factorization

Non-negative matrix factorization models based on a hierarchical Gamma-Poisson structure capture user and item behavior effectively in extremely sparse data sets, making them the ideal choice for collaborative filtering applications. Hierarchical Poisson factorization (HPF) in particular has proved successful for scalable recommendation systems with extreme sparsity. HPF, however, suffers from a tight coupling of sparsity model (absence of a rating) and response model (the value of the rating), which limits the expressiveness of the latter. Here, we introduce hierarchical compound Poisson factorization (HCPF) that has the favorable Gamma-Poisson structure and scalability of HPF to high-dimensional extremely sparse matrices. More importantly, HCPF decouples the sparsity model from the response model, allowing us to choose the most suitable distribution for the response. HCPF can capture binary, non-negative discrete, non-negative continuous, and zero-inflated continuous responses. We compare HCPF with HPF on nine discrete and three continuous data sets and conclude that HCPF captures the relationship between sparsity and response better than HPF.Comment: Will appear on Proceedings of the 33 rd International Conference on Machine Learning, New York, NY, USA, 2016. JMLR: W&CP volume 4

Expandable Factor Analysis

Bayesian sparse factor models have proven useful for characterizing dependence in multivariate data, but scaling computation to large numbers of samples and dimensions is problematic. We propose expandable factor analysis for scalable inference in factor models when the number of factors is unknown. The method relies on a continuous shrinkage prior for efficient maximum a posteriori estimation of a low-rank and sparse loadings matrix. The structure of the prior leads to an estimation algorithm that accommodates uncertainty in the number of factors. We propose an information criterion to select the hyperparameters of the prior. Expandable factor analysis has better false discovery rates and true positive rates than its competitors across diverse simulations. We apply the proposed approach to a gene expression study of aging in mice, illustrating superior results relative to four competing methods.Comment: 28 pages, 4 figure