Generalized Least Squares Kernelized Tensor Factorization

Abstract

Recovering incomplete multidimensional tensor-structured data is a fundamental task in many real-world applications. Smoothness-constrained low-rank tensor factorization effectively captures global and long-range correlations, but often struggles to characterize short-scale, high-frequency, or locally varying structures. We propose GLSKF, a complementary Generalized Least Squares Kernelized Tensor Factorization framework, for multidimensional spatiotemporal data completion. GLSKF additively integrates a covariance-regularized low-rank global component with an explicitly modeled locally correlated residual component under a GLS objective, enabling effective modeling of both global dependencies and localized variations. A covariance norm regularizer encodes spatiotemporal dependencies in both components: structured covariances are imposed on the latent factor columns to enforce smoothness in the global factorization, whereas compactly supported sparse kernels are used to model local correlations in the residual. We develop an alternating least squares algorithm with blockwise linear-system updates that exploit the Kronecker structure of the covariance matrices under missing data and facilitate fast conjugate gradient solves. Additional computational gains are obtained by exploiting the sparsity and Toeplitz structure of the local residual covariance matrices for efficient matrix-vector multiplications. We evaluate GLSKF on four real-world multidimensional data-completion tasks: traffic speed imputation, color image completion, digital video recovery, and MRI data reconstruction. Experimental results demonstrate that GLSKF achieves superior reconstruction performance and favorable scalability across a range of tensor completion tasks, supporting its broad applicability to multidimensional data completion.