The paper introduces a new semi‑tensor product for third‑order tensors that relaxes the dimensional constraints of the standard t‑product while preserving the closed‑form nature of T‑SVD. It builds a multi‑term semi‑tensor product singular value decomposition (MSTP‑SVD) that improves low‑rank approximation accuracy, and further accelerates it with randomized projection and power iteration to create the MRSTP‑SVD algorithm. Experiments on image and video compression and completion show that this method balances reconstruction accuracy and computational efficiency.
arXiv:2609.14307v1 Announce Type: new Abstract: Low-rank tensor factorization provides a flexible framework for completing multidimensional data from incomplete and corrupted observations. However, u...
arXiv:2606. 31061v1 Announce Type: cross Abstract: Tensor Train (TT) decomposition is a powerful technique for analyzing high-dimensional data.
arXiv:2609.12843v1 Announce Type: new Abstract: Recently, tensor decompositions are prevalent for multi-dimensional image representation, which learn the instance-specific structure of each image fro...
arXiv:2412. 07041v4 Announce Type: replace-cross Abstract: Recovering incomplete multidimensional tensor-structured data is a fundamental task in many real-world applications.
arXiv:2607. 25967v2 Announce Type: replace-cross Abstract: Singular Value Decomposition (SVD) underlies matrix factorisation tasks across many fields, with imaging applications demanding real-time processing.
The paper introduces TUON, a tensor‑aware optimizer that leverages Tensor‑SVD and a unitary transform to mix and polarize layers, thereby capturing cross‑layer correlations while preserving per‑layer diversity. Unlike previous matrix‑aware optimizers such as Muon, TUON avoids structural imbalance and misalignment caused by unfolding, enabling scaling to more layers and diverse modules beyond QKV. The authors prove that TUON performs steepest descent under a transformed tensor spectral norm, establish theoretical properties relative to Muon and TEON, and demonstrate empirical improvements on GPT‑124M/350M and Qwen‑0.6B/4B models.
The paper introduces DiffTR, an unsupervised framework that unifies tensor completion, denoising, and hyperspectral super‑resolution using coordinate‑conditioned diffusion. Shared factors indexed by tensor coordinates condition a compact score network, allowing the model to learn both the factors and an implicit tensor representation from observations. The framework separates representation learning from task‑specific observation modeling, enabling reverse diffusion for missing‑value recovery, Tweedie‑based denoising, and measurement‑guided sampling for high‑resolution reconstruction, all while supporting multiple tensor modalities and corruption models.
Matricized low-rank approximation via SVD is a standard surrogate for tensor decompositions, but entry-wise reconstruction error fails to capture multiway geometric degradation. Under an orthogonal Tucker model, we characterize this degradation using two metrics: cross-mode Direction Loss, measuring geometric subspace deviation from rank truncation and noise rotation, and Interaction Loss, quantifying multilinear interaction distortion in the core tensor.
arXiv:2606. 24989v1 Announce Type: new Abstract: Urban flow and air-quality simulations generate high-dimensional datasets describing velocity and pollutant transport across multiple spatial, temporal, and physical-variable dimensions.
arXiv:2606. 10085v1 Announce Type: new Abstract: Matrix-valued time series arise in a wide range of applications, such as spatio-temporal data from medical imaging and geophysics.
arXiv:2603.02720v2 Announce Type: replace Abstract: Recently, tensor decompositions have attracted increasing attention. Fundamentally, different interactions among factors induce distinct tensor dec...
The paper introduces Coupled Tensor‑Tensor Completion (CTTC), a new framework that incorporates side information in tensor form to enhance tensor completion tasks. CTTC leverages hidden connections among multimodal tensors and is grounded in distance metric learning and group theory. Experiments on the DTD and LINCS datasets show that CTTC outperforms existing methods such as HaLRTC, CTRC, Cell, and NTDDR in both run‑time and root‑sum‑of‑errors accuracy for predicting drug effects.