A New Tensor Network: Tubal Tensor Train and Its Applications

Abstract

We introduce the tubal tensor train (TTT) decomposition, a tensor-network model that combines the t-product algebra of the tensor singular value decomposition (T-SVD) with the low-order core structure of the tensor train (TT) format. For an order-$(N+1)$ tensor with a distinguished tube mode, the proposed representation consists of two third-order boundary cores and $N-2$ fourth-order interior cores linked through the t-product. As a result, for bounded tubal ranks, the storage scales linearly with the number of modes, in contrast to direct high-order extensions of T-SVD. We present two computational strategies: a sequential fixed-rank construction, called TTT-SVD, and a Fourier-slice alternating scheme based on the alternating two-cores update (ATCU). We also state a TT-SVD-type error bound for TTT-SVD and illustrate the practical performance of the proposed model on image compression, video compression, tensor completion, and hyperspectral imaging.

Figures (14)

• Figure 1: T-SVD and truncated T-SVD for a third-order tensor.
• Figure 2: Tubal tensor train structure and tube-wise evaluation.
• Figure 3: Benchmark images used in our experiments.
• Figure 4: The reconstructed images using the proposed TTT model and the TT-based model for an upper error bound of $0.15$.
• Figure 5: The PSNR and SSIM of the reconstructed frames by the proposed TTT model and TT decomposition for the relative approximation error bound of $0.1$ for the "Akiyo" video (left) and the "News" video datasets.

Theorems & Definitions (13)

• Definition 1: t-product kilmer2011factorization
• Definition 2: Transpose
• Definition 3: Identity tensor
• Definition 4: Orthogonal tensor
• Definition 5: T-SVD
• Definition 6: Tubal outer product
• Remark 1
• Theorem 1
• proof
• Example 1