提出新型张量分解方法，高效压缩图像视频等高维数据。
A New Tensor Network: Tubal Tensor Train and Its Applications
- 结合t-积代数与张量列车结构，构建新张量网络模型。
- 存储复杂度线性增长，支持高阶数据压缩与补全任务。
- 适用于图像、视频、遥感影像等多模态数据处理场景。
我们提出一种新的张量网络模型——管状张量列车（Tubal Tensor Train, TTT）分解，该方法将张量奇异值分解（T-SVD）中的t-积代数与张量列车（TT）格式的低阶核心结构相结合。针对具有特定管模式的(N+1)阶张量，该表示由两个三阶边界核心和N-2个四阶内部核心通过t-积连接而成。在管秩有界的情况下，存储空间随模式数线性增长，优于直接扩展的T-SVD方法。本文提出了两种计算策略：一种是顺序固定秩构造法（TTT-SVD），另一种是基于傅里叶切片交替更新的方案（ATCU）。同时给出了类似TT-SVD的误差上界，并在图像压缩、视频压缩、张量补全和高光谱成像任务中验证了其实际性能。
原文摘要 · Abstract (English)
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.
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