Problem
Research questions and friction points this paper is trying to address.
t-product
semi-tensor product
approximation accuracy
computational cost
high-dimensional visual data
Semi-Tensor Product-Based Multi-Term Randomized T-SVD and Its Visual Applications
🤖 AI Summary
本文提出了一种基于半张量积的多项随机T-SVD方法，解决了高维视觉数据处理中的维度兼容性和近似精度问题，同时通过随机投影和幂迭代技术降低了计算成本。
📝 Abstract
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.
Innovation
Methods, ideas, or system contributions that make the work stand out.
semi-tensor product
multi-term SVD
randomized projection
power iteration
high-dimensional data processing
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