Recent advances in IoT and biometric sensing technologies have led to the generation of massive and high-dimensional tensor data, yet achieving accurate and efficient low-rank approximation remains a major challenge. Most existing tensor decomposition methods require predefined ranks and iterative optimization, resulting in high computational costs and dependence on analyst expertise. This study proposes a novel tensor low-rank approximation method that eliminates both prior rank specification and iterative optimization. The method applies statistical singular value hard thresholding to each mode-wise unfolded matrix to automatically extract statistically significant components, effectively reducing noise while preserving the intrinsic structure. Theoretically, the optimal thresholds for each mode are derived from the asymptotic properties of the Marcenko-Pastur distribution. Simulation experiments demonstrate that the proposed method outperforms conventional approaches (HOSVD, HOOI, and Tucker-L2E) in both estimation accuracy and computational efficiency. These results indicate that the proposed approach provides a theoretically grounded, fully automatic, and non-iterative framework for tensor decomposition.

The application of low-order decomposition results in considerable memory savings while preserving the features essential for learning, and also offers theoretical guarantees to convergence.

Tensors provide a robust framework for managing high-dimensional data. Consequently, tensor analysis has emerged as an active research area in various domains, including machine learning, signal processing, computer vision, graph analysis, and data mining. This study introduces an efficient image storage approach utilizing tensors, aiming to minimize memory to store, bandwidth to transmit and energy to processing. The proposed method organizes original data into a higher-order tensor and applies the Tucker model for compression. Implemented in R, this method is compared to a baseline algorithm. The evaluation focuses on efficient of algorithm measured in term of computational time and the quality of information preserved, using both simulated and real datasets. A detailed analysis of the results is conducted, employing established quantitative metrics, with significant attention paid to sustainability in terms of energy consumption across algorithms.

We prove that the classic approximation guarantee for the higher-order singular value decomposition (HOSVD) is tight by constructing a tensor for which HOSVD achieves an approximation ratio of $N/(1+\varepsilon)$, for any $\varepsilon > 0$. This matches the upper bound of De Lathauwer et al. (2000a) and shows that the approximation ratio of HOSVD cannot be improved. Using a more advanced construction, we also prove that the approximation guarantees for the ST-HOSVD algorithm of Vannieuwenhoven et al. (2012) and higher-order orthogonal iteration (HOOI) of De Lathauwer et al. (2000b) are tight by showing that they can achieve their worst-case approximation ratio of $N / (1 + \varepsilon)$, for any $\varepsilon > 0$.

On-device learning has emerged as a promising direction for AI development, particularly because of its potential to reduce latency issues and mitigate privacy risks associated with device-server communication, while improving energy efficiency. Despite these advantages, significant memory and computational constraints still represent major challenges for its deployment. Drawing on previous studies on low-rank decomposition methods that address activation memory bottlenecks in backpropagation, we propose a novel shortcut approach as an alternative. Our analysis and experiments demonstrate that our method can reduce activation memory usage, even up to $120.09\times$ compared to vanilla training, while also reducing overall training FLOPs up to $1.86\times$ when evaluated on traditional benchmarks.

Strength: --The paper provides the theoretical analysis of approximation guarantees and a generalization bound for the class of tensor-valued regression functions. Weakness: --A major drawback is that the novelty and contribution is rather limited. The key idea and the model of this paper is actually equivalent to the HOPLS in the following paper: [Zhao et. In HOPLS, it assumes the tensor input has low-rank structure and also the tensor output has low-rank structure, and the link of them is established in the common latent space. And then follows a regression step against the projected latent variables.

Internet of Things and Deep Learning are synergetically and exponentially growing industrial fields with a massive call for their unification into a common framework called Edge AI. While on-device inference is a well-explored topic in recent research, backpropagation remains an open challenge due to its prohibitive computational and memory costs compared to the extreme resource constraints of embedded devices. Drawing on tensor decomposition research, we tackle the main bottleneck of backpropagation, namely the memory footprint of activation map storage. We investigate and compare the effects of activation compression using Singular Value Decomposition and its tensor variant, High-Order Singular Value Decomposition. The application of low-order decomposition results in considerable memory savings while preserving the features essential for learning, and also offers theoretical guarantees to convergence. Experimental results obtained on main-stream architectures and tasks demonstrate Pareto-superiority over other state-of-the-art solutions, in terms of the trade-off between generalization and memory footprint.

Emphasis in the tensor literature on random embeddings (tools for low-distortion dimension reduction) for the canonical polyadic (CP) tensor decomposition has left analogous results for the more expressive Tucker decomposition comparatively lacking. This work establishes general Johnson-Lindenstrauss (JL) type guarantees for the estimation of Tucker decompositions when an oblivious random embedding is applied along each mode. When these embeddings are drawn from a JL-optimal family, the decomposition can be estimated within $\varepsilon$ relative error under restrictions on the embedding dimension that are in line with recent CP results. We implement a higher-order orthogonal iteration (HOOI) decomposition algorithm with random embeddings to demonstrate the practical benefits of this approach and its potential to improve the accessibility of otherwise prohibitive tensor analyses. On moderately large face image and fMRI neuroimaging datasets, empirical results show that substantial dimension reduction is possible with minimal increase in reconstruction error relative to traditional HOOI ($\leq$5% larger error, 50%-60% lower computation time for large models with 50% dimension reduction along each mode). Especially for large tensors, our method outperforms traditional higher-order singular value decomposition (HOSVD) and recently proposed TensorSketch methods.

This work studies the combinatorial optimization problem of finding an optimal core tensor shape, also called multilinear rank, for a size-constrained Tucker decomposition. We give an algorithm with provable approximation guarantees for its reconstruction error via connections to higher-order singular values. Specifically, we introduce a novel Tucker packing problem, which we prove is NP-hard, and give a polynomial-time approximation scheme based on a reduction to the 2-dimensional knapsack problem with a matroid constraint. We also generalize our techniques to tree tensor network decompositions. We implement our algorithm using an integer programming solver, and show that its solution quality is competitive with (and sometimes better than) the greedy algorithm that uses the true Tucker decomposition loss at each step, while also running up to 1000x faster.