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Life sciences · Preprint
arXiv · September 10, 2026
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This preprint presents a novel semi-tensor product mathematical framework and a multi-term randomized singular value decomposition algorithm for high-dimensional tensor data. It is a theoretical contribution with proof-of-concept computational experiments on image and video compression and completion, not a clinical study or validated medical intervention.
Preprint.
MSTP-SVD integrates multiple orthogonal decomposition terms to improve low-rank approximation accuracy versus single-term schemes MRSTP-SVD incorporates randomized projection and power iteration to balance reconstruction accuracy and computational efficiency Novel semi-tensor product construction breaks dimensional matching constraints of standard t-product while retaining closed-form T-SVD properties
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This is a mathematics/computer science methods paper presenting novel algorithmic theory without clinical validation, application to patient outcomes, or peer review.
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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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