Bayesian Factorizations of Big Sparse Tensors

• 成果类型: Article
• 署名作者: Zhou, Jing; Bhattacharya, Anirban; Herring, Amy H.; Dunson, David B.
• 署名单位: University of North Carolina; University of North Carolina Chapel Hill; Texas A&M University System; Texas A&M University College Station; University of North Carolina; University of North Carolina Chapel Hill; Duke University
• 刊物名称: JOURNAL OF THE AMERICAN STATISTICAL ASSOCIATION
• ISSN/ISSBN: 0162-1459
• DOI: 10.1080/01621459.2014.983233
• 发表日期: 2015
• 页码: 1562-1576
• 关键词: posterior distributions; linear-models; Asymptotic Normality; variable-selection; convergence-rates; regression; Consistency; contraction; shrinkage; inference

摘要

It has become routine to collect data that are structured as multiway arrays (tensors). There is an enormous literature on low rank and sparse matrix factorizations, but limited consideration of extensions to the tensor case in statistics. The most common low rank tensor factorization relies on parallel factor analysis (PARAFAC), which expresses a rank k tensor as a sum of rank one tensors. In contingency table applications in which the sample size is massively less than the number of cells in the table, the low rank assumption is not sufficient and PARAFAC has poor performance. We induce an additional layer of dimension reduction by allowing the effective rank to vary across dimensions of the table. Taking a Bayesian approach, we place priors on terms in the factorization and develop an efficient Gibbs sampler for posterior computation. Theory is provided showing posterior concentration rates in high-dimensional settings, and the methods are shown to have excellent performance in simulations and several real data applications.