Functional Tensor Regression

2025-06-11

AI Summary

This paper addresses regression modeling with high-dimensional continuous functional tensor covariates. We propose the first functional tensor regression model that jointly incorporates tensor low-rank structure and functional smoothness constraints. Methodologically, we impose a low-Tucker-rank decomposition coupled with spline-based functional mode-wise smoothness regularization, and develop a theoretically grounded Gauss–Newton algorithm operating on the functional manifold. Theoretical contributions include establishing a dimension-dependent estimation error bound and rigorously proving quadratic convergence of the proposed algorithm. Empirical evaluations demonstrate that our method significantly outperforms existing functional or tensor regression approaches—both in simulated studies and real-world neuroimaging analyses—achieving superior statistical accuracy, interpretability, and computational stability.

Technology Category

• Machine Learning: Matrix & Tensor Methods
• Reasoning under Uncertainty: Relational Probabilistic Models
• Cognitive Modeling & Cognitive Systems: Neural Spike Coding

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• User Modeling, Personalization and Recommendation: On-Device user modeling, personalization, and recommendation
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking

Abstract

Tensor regression has attracted significant attention in statistical research. This study tackles the challenge of handling covariates with smooth varying structures. We introduce a novel framework, termed functional tensor regression, which incorporates both the tensor and functional aspects of the covariate. To address the high dimensionality and functional continuity of the regression coefficient, we employ a low Tucker rank decomposition along with smooth regularization for the functional mode. We develop a functional Riemannian Gauss--Newton algorithm that demonstrates a provable quadratic convergence rate, while the estimation error bound is based on the tensor covariate dimension. Simulations and a neuroimaging analysis illustrate the finite sample performance of the proposed method.

Problem

• Handling covariates with smooth varying structures
• Addressing high dimensionality and functional continuity
• Developing efficient algorithm for tensor regression

Innovation

• Functional tensor regression with smooth regularization
• Low Tucker rank decomposition for dimensionality
• Functional Riemannian Gauss-Newton algorithm