Generalized Least Squares Kernelized Tensor Factorization
Mengying Lei, Lijun Sun
TL;DR
GLSKF addresses the challenge of completing multidimensional tensors with missing entries by uniting a smooth, kernelized global low-rank factorization with a locally correlated residual component. The framework introduces a generalized covariance (K-norm) to enforce smoothness, and solves the resulting objective via alternating least squares with closed-form subproblems, exploiting Kronecker structure and projection-based missing-data handling for efficiency. Empirical results across traffic speed imputation, color image inpainting, color video completion, and MRI reconstruction show that GLSKF outperforms state-of-the-art low-rank methods in both accuracy (lower MAE/RMSE and higher PSNR/SSIM) and scalability, often with lower tensor ranks. This additive global-local modeling yields high-quality reconstructions while remaining computationally efficient for large-scale tensor data, making it suitable for real-world spatiotemporal applications.
Abstract
Completing multidimensional tensor-structured data with missing entries is a fundamental task for many real-world applications involving incomplete or corrupted datasets. For data with spatial or temporal side information, low-rank factorization models with smoothness constraints have demonstrated strong performance. Although effective at capturing global and long-range correlations, these models often struggle to capture short-scale, high-frequency variations in the data. To address this limitation, we propose the Generalized Least Squares Kernelized Tensor Factorization (GLSKF) framework for tensor completion. GLSKF integrates smoothness-constrained low-rank factorization with a locally correlated residual process; the resulting additive structure enables effective characterization of both global dependencies and local variations. Specifically, we define the covariance norm to enforce the smoothness of factor matrices in the global low-rank factorization, and use structured covariance/kernel functions to model the local processes. For model estimation, we develop an alternating least squares (ALS) procedure with closed-form solutions for each subproblem. GLSKF utilizes zero-padding and slicing operations based on projection matrices which preserve the Kronecker structure of covariances, facilitating efficient computations through the conjugate gradient (CG) method. The proposed framework is evaluated on four real-world datasets across diverse tasks. Experimental results demonstrate that GLSKF achieves superior performance and scalability, establishing it as a novel solution for multidimensional tensor completion.