Online TT-ALS for Streaming Tensor Decomposition with Incremental Orthogonalization

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.

We propose a scalable tensorization framework for neural network compression based on slice-wise feature distillation. Unlike conventional tensor decomposition methods that rely on costly global finetuning, our approach decomposes the network into slices consisting of either individual layers or blocks (e.g., convolutional layers or MLPs), or small groups of consecutive layers, and tensorizes each slice independently to reproduce the intermediate representations of the original pretrained model. This modular strategy improves accuracy recovery, reduces data requirements, and enables efficient parallel optimization. Experiments on ResNet-34 show significant gains over conventional global tensorization, achieving near-lossless compression at moderate compression rates with faster optimization. Results on GPT-2 XL further demonstrate the scalability of the method and its applicability to large-scale models, particularly in distributed settings.

We consider a novel algorithm, for the completion of partially observed low-rank tensors, as a generalization of matrix completion. The proposed low-rank tensor completion (TC) method builds on the conventional nuclear norm (NN) minimization-based low-rank TC paradigm, by leveraging the alternating direction method of multipliers (ADMM) optimization framework. To that extend the original NN minimization problem is reformulated into multiple subproblems, which are then solved iteratively via closed-form proximal operators, making use of over-relaxation and an adaptive penalty parameter update scheme, to further speed up convergence and improve the overall performance of the method. Simulation results demonstrate the superior performance of the new method in terms of normalized mean square error (NMSE), compared to the conventional state-of-the-art (SotA) techniques, including NN minimization approaches, as well as a mixture of the latter with a matrix factorization approach, while its convergence can be significantly improved by initializing the algorithm with the solution of the SotA.