Semi-Tensor Product-Based Multi-Term Randomized T-SVD and Its Visual Applications

• URL: http://arxiv.org/abs/2609.11168v1
• Date: Thu, 10 Sep 2026 07:14:09 GMT
• Title: Semi-Tensor Product-Based Multi-Term Randomized T-SVD and Its Visual Applications
• Abstract summary: This paper introduces a novel semi-tensor product for third-order tensors under the t-product framework induced by arbitrary invertible linear transforms.<n>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.
• Score: 10.173625999044658
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• 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.

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