Regularized Tensor Factorizations and Higher-Order PCA

• Activity Number: 587
• Type: Invited
• Date/Time: Thursday, August 2, 2012 : 8:30 AM to 10:20 AM
• Sponsor: Section on Statistical Computing
• Author(s): Genevera Allen*+
• Companies: Rice University
• Keywords: tensor decompositions ; principal components analysis ; sparse principal components analysis ; functional principal components anaylsis ; Tucker decomposition ; CP decomposition

Abstract

High-dimensional tensors or multi-way data are becoming prevalent in areas such as biomedical imaging, chemometrics, networking and bibliometrics. Traditional approaches to finding lower dimensional representations of tensor data include flattening the data and applying matrix factorizations or employing tensor decompositions such as the CANDECOMP / PARAFAC (CP) and Tucker decompositions. The former can lose important structure in the data, while the latter can be problematic in high-dimensions with many irrelevant features. We introduce frameworks for regularized tensor factorizations based on heuristic algorithmic approaches and by solving penalized optimization problems related to the CP and Tucker decompositions. These approaches lead to methods for Sparse Higher-Order PCA and Functional Higher-Order PCA. We illustrate the utility of our methods for dimension reduction, feature selection, and signal recovery on simulated data and multi-dimensional microarrays and spectroscopy.