Problem
Research questions and friction points this paper is trying to address.
Poisson regression
tensor-on-tensor regression
count data
multi-dimensional arrays
structured data
Poisson-response Tensor-on-Tensor Regression and Applications
🤖 AI Summary
This study addresses the challenge of regression modeling between multidimensional Poisson count responses and tensor-valued covariates by proposing a Poisson Tensor-on-Tensor Regression (PToTR) framework. The method introduces a canonical polyadic (CP) decomposition structure on the regression coefficient tensor, thereby incorporating the Poisson response mechanism into tensor-on-tensor regression for the first time. This formulation naturally enforces non-negativity of parameters and enables an efficient optimization algorithm based on maximum likelihood estimation. Theoretical analysis establishes error bounds for the resulting estimator, extending the modeling paradigm for high-dimensional count data. Empirical evaluations demonstrate the method’s superior performance and broad applicability across diverse tasks, including crisis forecasting, PET image reconstruction, and change-point detection in dynamic communication networks.
📝 Abstract
We introduce Poisson-response tensor-on-tensor regression (PToTR), a novel regression framework designed to handle tensor responses composed element-wise of random Poisson-distributed counts. Tensors, or multi-dimensional arrays, composed of counts are common data in fields such as international relations, social networks, epidemiology, and medical imaging, where events occur across multiple dimensions like time, location, and dyads. PToTR accommodates such tensor responses alongside tensor covariates, providing a versatile tool for multi-dimensional data analysis. We propose algorithms for maximum likelihood estimation under a canonical polyadic (CP) structure on the regression coefficient tensor that satisfy the positivity of Poisson parameters and then provide an initial theoretical error analysis for PToTR estimators. We also demonstrate the utility of PToTR through three concrete applications: longitudinal data analysis of the Integrated Crisis Early Warning System database, positron emission tomography (PET) image reconstruction, and change-point detection of communication patterns in longitudinal dyadic data. These applications highlight the versatility of PToTR in addressing complex, structured count data across various domains.
Innovation
Methods, ideas, or system contributions that make the work stand out.
Poisson-response tensor regression
tensor-on-tensor regression
canonical polyadic decomposition
count data
maximum likelihood estimation
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