A0939
Title: Generalized multilinear models for tensor-on-tensor regression
Authors: Carlos Llosa - Sandia National Laboratories (United States) [presenting]
Daniel M Dunlavy - Sandia National Laboratories (United States)
Richard B Lehoucq - Sandia National Laboratories (United States)
Jeremy Myers - Sandia National Laboratories (United States)
Tian Ma - Sandia National Laboratories (United States)
Abstract: The generalized multilinear model (GMLM) is introduced, a novel modeling framework that extends the generalized linear model (GLM) and tensor-on-tensor regression (ToTR) for regression problems involving tensor, or multidimensional array, data. As with GLMs, GMLMs allow a linear model to relate expected response variables following arbitrary distributions to covariates via a general link function, providing flexibility in solving problems beyond typical identity link/Gaussian-response regression. As in ToTR, GMLMs allow for tensor covariates and responses, providing models that can leverage the multilinear structure inherent in many data that is often discarded when the data is vectorized and modeled entry-wise using scalar-response GLMs. Vectorizing the data often leads to an ill-posed problem unless provided a large sample size that increases with the product of the sizes of the covariate and response tensors. Instead, a low-rank tensor structure is imposed on the GMLM parameter tensor, thus requiring fewer samples and leading to a well-posed inference problem. The extensions of GLMs and ToTR that lead to GMLMs are discussed, an algorithmic framework is introduced for solving the GMLM parameter inference problem when the low-rank structure imposed on the parameter tensor is the Canonical Polyadic (CP) model, and multiple uses of GMLMs are illustrated on simulated and real-world application data.
