Accelerated Tensor Completion via Trace-Regularized Fully-Connected Tensor Network

TL;DR

This work tackles the challenge of recovering local details in high-dimensional tensor completion by integrating trace regularization into the fully-connected tensor network ($\mathrm{FCTN}$) framework. The proposed AFCTNLR model leverages trace penalties on mode-$k$ unfoldings with a periodically modified negative Laplacian to promote smoothness of the FCTN factors, improving local detail recovery without sacrificing global low-rank structure. A proximal alternating minimization (PAM) algorithm solves the model, and an intermediate tensor reutilization mechanism reduces runtime by 10%–30% while preserving accuracy; convergence to a critical point is theoretically established under the K-Ł property. Empirical results on color videos and multi-temporal hyperspectral images show that AFCTNLR yields superior PSNR/SSIM and competitive runtimes compared with state-of-the-art baselines, demonstrating practical impact for high-dimensional tensor completion tasks.

Abstract

The fully-connected tensor network (FCTN) decomposition has gained prominence in the field of tensor completion owing to its powerful capacity to capture the low-rank characteristics of tensors. Nevertheless, the recovery of local details in the reconstructed tensor still leaves scope for enhancement. In this paper, we propose efficient tensor completion model that incorporates trace regularization within the FCTN decomposition framework. The trace regularization is constructed based on the mode-$k$ unfolding of the FCTN factors combined with periodically modified negative laplacian. The trace regularization promotes the smoothness of the FCTN factors through discrete second-order derivative penalties, thereby enhancing the continuity and local recovery performance of the reconstructed tensor. To solve the proposed model, we develop an efficient algorithm within the proximal alternating minimization (PAM) framework and theoretically prove its convergence. To reduce the runtime of the proposed algorithm, we design an intermediate tensor reuse mechanism that can decrease runtime by 10\%-30\% without affecting image recovery, with more significant improvements for larger-scale data. A comprehensive complexity analysis reveals that the mechanism attains a reduced computational complexity. Numerical experiments demonstrate that the proposed method outperforms existing approaches.

Figures (9)

• Figure 1: The fourth-order FCTN decomposition of tensor.
• Figure 2: Common intermediate tensor of $\mathcal{M}_1$ and $\mathcal{M}_2$.
• Figure 3: Common intermediate tensor of $\mathcal{M}_3$ and $\mathcal{M}_4$.
• Figure 4: Generate $\operatorname{FCTN}\left(\left\{\mathcal{A}_{k}^{(t+1)}\right\}_{k=1}^{4}\right)$ using $\mathcal{M}_{4}^{(t)}$.
• Figure 5: Distinct combinations for generating $\operatorname{FCTN}\left(\left\{\mathcal{A}_{k}^{(t+1)}\right\}_{k=1}^{4}\right)$.

Theorems & Definitions (3)

• Theorem 1
• Theorem 2
• proof