基于半张量积的多项随机T-SVD及其视觉应用
Semi-Tensor Product-Based Multi-Term Randomized T-SVD and Its Visual Applications
- School of Mathematics and Statistics, Southwest University（西南大学数学与统计学院）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
针对标准t-product维度限制和单项STP近似精度不足的问题，本文提出基于半张量积的多项随机T-SVD（MRSTP-SVD），通过多项分解和随机化技术提升低秩近似精度与计算效率，并在图像视频压缩补全任务中验证有效性。
AI中文摘要：
张量奇异值分解（T-SVD）建立在张量-张量积（t-product）之上，已成为处理彩色图像和视频等高维视觉数据的强大工具。然而，标准的t-product对维度兼容性施加了严格的限制。尽管基于半张量积（STP）的扩展放宽了这一限制，但其单项形式仍然面临近似精度有限的问题。此外，这些确定性方法在处理大规模张量数据时计算成本高昂。为解决这些问题，本文在由任意可逆线性变换诱导的t-product框架下，引入了一种新的三阶张量半张量积。由此产生的张量半张量积打破了标准t-product的刚性维度匹配要求，同时保留了T-SVD的闭式性质。基于这一构造，我们开发了多项半张量积奇异值分解（MSTP-SVD），它整合了多个正交分解项，与单项方案相比显著提高了低秩近似精度。为降低多项建模的计算成本，我们将随机投影和幂迭代技术纳入MSTP-SVD框架，提出了一种加速的多项随机半张量积SVD（MRSTP-SVD）算法，该算法在重建精度和计算效率之间取得了平衡。在图像和视频压缩及补全任务上的实验证明了所提方法的有效性。
英文摘要：
Tensor singular value decomposition (T-SVD), which is built upon the tensor-tensor product (t-product), has emerged as a powerful tool for processing high-dimensional visual data such as color images and videos. However, the standard t-product imposes strict dimensional compatibility constraints. Although extensions based on the semi-tensor product (STP) relax this restriction, their single-term formulations still suffer from limited approximation accuracy. Moreover, these deterministic methods incur high computational costs when processing large-scale tensor data. To address these issues, this paper introduces a novel semi-tensor product for third-order tensors under the t-product framework induced by arbitrary invertible linear transforms. The resulting tensor semi-tensor product breaks the rigid dimension matching requirement of the standard t-product, while retaining the closed-form property of T-SVD. Based on this construction, we develop a multi-term semi-tensor product singular value decomposition (MSTP-SVD), which integrates multiple orthogonal decomposition terms to significantly improve low-rank approximation accuracy compared with single-term schemes. To reduce the computational cost of multi-term modeling, we incorporate randomized projection and power iteration techniques into the MSTP-SVD framework, yielding an accelerated multi-term randomized semi-tensor product SVD (MRSTP-SVD) algorithm that achieves a balance between reconstruction accuracy and computational efficiency. Experiments on image and video compression and completion tasks demonstrate the effectiveness of the proposed method.
