Fourier Low-rank and Sparse Tensor for Efficient Tensor Completion
Jingyang Li, Jiuqian Shang, Yang Chen
TL;DR
This paper addresses tensor completion for spatiotemporal data by moving beyond symmetric low-rank models to a Fourier-domain hybrid: low-frequency components are modeled as low-rank and high-frequency components as sparse, forming the Fourier Low-rank and Sparse Tensor (FLoST). The authors derive an efficient, parallelizable estimator that decouples into independent subproblems in the Fourier domain, employs singular value shrinkage for low-rank parts and soft-thresholding for sparse parts, and provide a provable, near-optimal error bound that can be independent of the time dimension. Theoretical guarantees show the estimator achieves favorable sample complexity with dof scaling and improves upon tubal-rank bounds, while numerical experiments on synthetic data and TEC maps demonstrate higher accuracy and lower computational cost compared to baselines. Overall, FLoST offers a physically aligned, scalable approach for reconstructing large spatiotemporal tensors with missing data, delivering interpretable structure and practical impact for TEC and related domains.
Abstract
Tensor completion is crucial in many scientific domains with missing data problems. Traditional low-rank tensor models, including CP, Tucker, and Tensor-Train, exploit low-dimensional structures to recover missing data. However, these methods often treat all tensor modes symmetrically, failing to capture the unique spatiotemporal patterns inherent in scientific data, where the temporal component exhibits both low-frequency stability and high-frequency variations. To address this, we propose a novel model, \underline{F}ourier \underline{Lo}w-rank and \underline{S}parse \underline{T}ensor (FLoST), which decomposes the tensor along the temporal dimension using a Fourier transform. This approach captures low-frequency components with low-rank matrices and high-frequency fluctuations with sparsity, resulting in a hybrid structure that efficiently models both smooth and localized variations. Compared to the well-known tubal-rank model, which assumes low-rankness across all frequency components, FLoST requires significantly fewer parameters, making it computationally more efficient, particularly when the time dimension is large. Through theoretical analysis and empirical experiments, we demonstrate that FLoST outperforms existing tensor completion models in terms of both accuracy and computational efficiency, offering a more interpretable solution for spatiotemporal data reconstruction.