An Efficient Tensor Completion Method Via New Latent Nuclear Norm

Abstract

In tensor completion, the latent nuclear norm is commonly used to induce low-rank structure, while substantially failing to capture the global information due to the utilization of unbalanced unfolding schemes. To overcome this drawback, a new latent nuclear norm equipped with a more balanced unfolding scheme is defined for low-rank regularizer. Moreover, the new latent nuclear norm together with the Frank-Wolfe (FW) algorithm is developed as an efficient completion method by utilizing the sparsity structure of observed tensor. Specifically, both FW linear subproblem and line search only need to access the observed entries, by which we can instead maintain the sparse tensors and a set of small basis matrices during iteration. Most operations are based on sparse tensors, and the closed-form solution of FW linear subproblem can be obtained from rank-one SVD. We theoretically analyze the space-complexity and time-complexity of the proposed method, and show that it is much more efficient over other norm-based completion methods for higher-order tensors. Extensive experimental results of visual-data inpainting demonstrate that the proposed method is able to achieve state-of-the-art performance at smaller costs of time and space, which is very meaningful for the memory-limited equipment in practical applications. INDEX TERMS Tensor completion, tensor ring decomposition, tensor ring rank, latent nuclear norm, image/video inpainting.

Tensor Completion Algorithm and Its Applications to Wireless Edge Caching and Hyper- Spectral Imaging

“…• Simulations For wireless edge caching with the Movie-Lens dataset show lower reconstruction errors for the proposed algorithm than the recent FW algorithm, albeit with lower computation overhead. They also present the improvements in the normalized cache hit rate compared to conventional CP decomposition (CPD) [45], and the FW method [43], particularly at higher decomposition ranks. • To assess the versatility and practicality of our proposed tensor completion algorithm, we conducted simulations with Hyper-Spectral Image datasets, including the Pavia University and Pavia city datasets, with noisy data.…”

“…The random-mode FW algorithm introduced in our work builds upon the foundations laid by previous studies in latent-norm-based methods [41]- [43] and TR-based approaches [37], [40]. The study in [41] utilized the latent nuclear norm, while [42] innovated a new latent nuclear norm through the Tensor Train framework.…”

“…To enhance efficiency, [40] employed an overlapped TR nuclear norm, significantly reducing computational complexity per iteration. Additionally, [43] streamlined the process by proposing a novel latent TRnuclear norm, which effectively minimizes the number of parameters required for selection. Our work seeks to further these advancements by presenting an optimized randommode FW algorithm that integrates these concepts while addressing their limitations.…”

“…This approach overcomes the drawbacks of unbalanced unfolding and casts the tensor completion task as a convex optimization problem. By modifying the Frank-Wolfe (FW) algorithm [43] with cyclic unfolding and gradient descent updates, we achieve a solution with significantly reduced time complexity. Our algorithm's versatility and efficiency are demonstrated through simulations on real-world datasets.…”

“…In this novel tensor completion algorithm, inspired by the TR decomposition, we employ convex TR-based latent nuclear norm based on cyclic unfolding, i.e., overcoming the drawback of unbalanced unfolding. The tensor completion task can be cast as a convex optimization problem to minimize the latent norm, which is solved via a modified FW algorithm [43] with cyclic unfolding to provide an efficient solution towards lower time complexity. In this algorithm, the FW method is modified to obtain a lowtime complexity solution via gradient descent updates.…”