Search Results (126)

The solution theory of Sylvester-type equations finds wide applications in control theory, robotics, and image processing. This paper systematically surveys, classifies and summarizes the existing research results of three classes of Sylvester-type equations: matrix equations, tensor equations, and operator equations. It extracts nine mainstream research methods and clarifies the internal correlations among these methods, as well as their applicable equation types. Combined with four prior review articles focusing on special cases of Sylvester-type equations, this work establishes a comprehensive framework for solving such equations. It not only provides a systematic theoretical foundation and a clear research thread for subsequent researchers but also offers valuable methodological insights for further investigations in related fields. Full article

The recursive least-squares (RLS) algorithm is an appealing choice in many adaptive filtering applications, especially due to its fast convergence rate. In the context of echo cancellation, the implementation of the conventional RLS algorithm is a challenging task, mainly due to the long length of the impulse response to be identified, which corresponds to the echo path. The computational complexity of this algorithm is proportional to the square of the filter length, which becomes prohibitive since the lengths of such impulse responses could be on the order of hundreds/thousands of coefficients. Recently, an efficient solution has been developed, which exploits a fourth-order (tensor-based) decomposition of the filter impulse response, using the nearest Kronecker product. This approach fits very well for low-rank impulse responses, like in echo cancellation. The resulting RLS-type algorithm designed in this context combines four (much shorter) adaptive filters, which work in parallel and are connected via their coefficients. Thus, the system owns an intrinsic symmetry-related structure. The overall behavior of this decomposition-based RLS algorithm is controlled by the forgetting factors associated to the four component filters. In order to achieve a proper balance between the main performance criteria (e.g., tracking versus accuracy), the current paper proposes a version of this algorithm with variable forgetting factors (VFFs). The design of these VFFs results in a practical manner, without requiring other additional control mechanisms or extra tuning parameters. Extensive simulation results obtained in the framework of echo cancellation support the performance features of the proposed decomposition-based RLS-type algorithm with VFFs. Full article

Being capable of high-resolution imaging and non-contact measurement, Ground Penetrating Radar (GPR) is a promising technology for the detection of unexploded ordnance (UXO). However, UXO detection is severely hindered by clutter, particularly in environments with significant surface roughness where conventional suppression methods prove ineffective. To address this, we propose a polarimetric GPR clutter suppression method based on an improved non-convex Tensor Robust Principal Component Analysis (TRPCA) framework. Specifically, a polarization-aware tensor construction scheme is designed by stacking the HH and VV channel data. This approach exploits the strong inter-channel correlation of clutter to enhance its low-rank property, while highlighting the distinct sparse signatures of targets derived from their polarimetric responses. To further optimize tensor decomposition, we introduce a non-convex Tensor Adjustable Logarithmic Norm (TALN) to overcome the estimation bias inherent in the conventional Tensor Nuclear Norm (TNN). Serving as a tighter surrogate for tensor rank, the proposed TALN regularizer improves the approximation accuracy of the low-rank component, thereby ensuring a clearer separation between clutter and targets. The resulting non-convex optimization problem is efficiently solved using Alternating Direction Method of Multipliers (ADMM). Numerical simulations and laboratory experiments demonstrate that the proposed method suppresses strong clutter stemming from rough-surface reflections more effectively than existing methods, achieving a Signal-to-Clutter Ratio (SCR) improvement of over 20 dB. Full article

This paper extends the theory of equivalence canonical forms from quaternion matrices to quaternion tensors under the Einstein product. Motivated by recent results on the simultaneous decomposition of two specific configurations of five quaternion matrices, we establish a comprehensive framework for the corresponding configurations of five quaternion tensors. The core approach leverages bijective transformation maps that establish isomorphisms between quaternion tensor spaces and matrix spaces, allowing us to systematically construct invertible transformation tensors that simultaneously reduce the given tensor quintuples to canonical forms consisting solely of binary entries (0 and 1). A detailed structural analysis of the resulting canonical tensor forms is provided, including explicit dimension formulas for all identity blocks derived from precise rank conditions. To demonstrate practical utility, we integrate the proposed tensor decomposition with the discrete wavelet transform to construct a color video encryption and decryption system. Experimental results confirm perfect reconstruction (PSNR exceeding 300 dB, SSIM equal to 1) and strong security performance: NPCR of 49.8%, UACI of 49.6%, information entropy of 0.9986 bits per pixel, adjacent pixel correlation below 0.03 in absolute value, and a key space exceeding 2⁵¹². The developed theory significantly extends the existing literature on quaternion tensor decompositions and provides powerful tools for multidimensional signal processing. Full article

Deep learning has emerged as an effective tool for automatic waste classification, supporting cleaner cities and more sustainable recycling systems. Because environmental protection is central to the United Nations Sustainable Development Goals (SDGs), improving the sorting and processing of everyday waste is a practical step toward this broader objective. In many real-world settings, however, waste is still sorted manually, which is slow, labor-intensive, and prone to human error. Although convolutional neural networks (CNNs) can automate this task with high accuracy, many state-of-the-art models remain too large and computationally demanding for low-cost edge devices intended for deployment in homes, schools, and small recycling facilities. In this work, we investigate lightweight waste-classification models suitable for TinyML deployment while preserving competitive accuracy. We first benchmark multiple CNN architectures to establish a strong baseline, then apply complementary compression strategies including quantization, pruning, singular value decomposition (SVD) low-rank approximation, and knowledge distillation. In addition, we evaluate an RL-guided multi-teacher selection benchmark that adaptively chooses one teacher per minibatch during distillation to improve student training stability, achieving up to 85% accuracy with only 0.496 M parameters (FP32 ≈ 1.89 MB; INT8 ≈ 0.47 MB). Across all experiments, the best accuracy–size trade-off is obtained by combining knowledge distillation with post-training quantization, reducing the model footprint from approximately 16 MB to 281 KB while maintaining 82% accuracy. The resulting model is feasible for deployment on mobile applications and resource-constrained embedded devices based on model size and TensorFlow Lite Micro compatibility. Full article

To address the issue of missing position in flight trajectory data collected by Automatic Dependent Surveillance-Broadcast (ADS-B) systems, a flight trajectory tensor completion model based on truncated Schatten p-norm minimization is proposed. First, the low-rank characteristics of the trajectory set are validated using Singular Value Decomposition (SVD); based on this, the data is transformed into a three-dimensional tensor structure. Next, a regularization strategy combining the Schatten p-norm with a singular value truncation mechanism is introduced to construct the trajectory tensor completion model, which suppresses noise and interference from minor components while preserving the main variation patterns of the trajectories. Finally, the model is optimized and solved using the Alternating Direction Method of Multipliers (ADMM) to obtain the completed trajectories. Taking historical ADS-B trajectory data from Orly Airport to Toulouse Airport as an example, the completion results of the proposed model under different missing patterns, missing rates, and flight phases are analyzed from both qualitative and quantitative perspectives. Experimental results show that compared with other representative models, the proposed model achieves the best completion performance under different missing patterns and missing rates; the completion performance during the cruise phase is better than during the ascent and descent phases. The proposed model can serve as a preprocessing technique for flight trajectory data in air traffic, providing more complete and reliable data support for various downstream applications. Full article

Tensor completion is an extension of matrix completion aimed at recovering a multiway data tensor by leveraging a given subset of its entries (observations) and the pattern of observation. The low-rank assumption is key in establishing a relationship between the observed and unobserved entries of the tensor. The low-rank tensor completion problem is typically solved using numerical optimization techniques, where the rank information is used either implicitly (in the rank minimization approach) or explicitly (in the error minimization approach). Current theories concerning these techniques often study probabilistic recovery guarantees under conditions such as random uniform observations and incoherence requirements. However, if an observation pattern exhibits some low-rank structure that can be exploited, more efficient algorithms with deterministic recovery guarantees can be designed by leveraging this structure. This work shows how to use only standard linear algebra operations to compute the tensor train decomposition of a specific type of “fiber-wise” observed tensor, where some of the fibers of a tensor (along a single specific mode) are either fully observed or entirely missing, unlike the usual entry-wise observations. From an application viewpoint, this setting is relevant when it is easier to sample or collect a multiway data tensor along a specific mode (e.g., temporal). The proposed completion method is fast and is guaranteed to work under reasonable deterministic conditions on the observation pattern. Through numerical experiments, we showcase interesting applications and use cases that illustrate the effectiveness of the proposed approach. Full article

This paper presents an orthogonal-heading wavelength-resolution SAR (WRSAR) target detection framework that fuses multi-heading image stacks for foliage-penetrating (FOPEN) vehicle detection. First, a low-rank–sparse decomposition is applied to very-high-frequency (VHF), ultra-wideband (UWB) WRSAR stacks to suppress vegetation clutter and enhance target contrast. The clutter-suppressed sparse stacks acquired from orthogonal headings are then fused to enrich target scattering characteristics. Finally, a Rayleigh-entropy statistic computed on the fused sparse stack is used to represent discontinuous positional changes. Based on the non-negative nature of WRSAR amplitudes for both clutter and FOPEN targets, we introduce a non-negative constrained tensor robust principal component analysis (NCTRPCA) to improve sparsity in the stack components. Furthermore, since Shannon differential entropy has no tunable parameter, we replace Shannon entropy with RE in this work and derive its closed-form expression for the proposed detector. Experiments on the publicly available multi-heading, multi-temporal CARABAS II dataset show that the proposed orthogonal-heading WRSAR fusion achieves higher FOPEN vehicle detection performance than recent state-of-the-art methods while maintaining moderate computational cost. Full article

Adaptive filtering algorithms based on tensor decomposition represent appealing choices for system identification problems, especially when dealing with the estimation of long-length impulse responses, like in acoustic echo cancellation. The topic has recently been addressed in the literature, showing that the gain (compared to the conventional approach) is twofold in terms of both better performance and lower complexity. The main idea is that a system identification problem with a large parameter space (i.e., a long-length filter) is reformulated based on a group of shorter filters, while their coefficients are combined using the Kronecker product. Nevertheless, one of the main challenges is related to handling the tensor rank, which is particularly addressed for each specific decomposition order. Previous solutions have been designed for second-order (matrix case) and third-order tensorial decompositions. In this paper, we develop a recursive least-squares adaptive filtering algorithm that exploits a fourth-order tensor decomposition, aiming for further performance improvements compared to the existing solutions. In this framework, the influence of the decomposition setup is investigated, which is also related to the main parameters of the algorithm, i.e., the forgetting factors. Simulations performed in the context of acoustic echo cancellation support the theoretical findings and indicate the good performance of the proposed algorithm. Full article

Synthetic aperture radar (SAR) imaging is highly vulnerable to radio-frequency interference (RFI) in complex electromagnetic environments, which can introduce structured artifacts and obscure targets in single-look complex (SLC) products. Most existing suppression methods rely on separability along a single dimension or require interference-specific parameter tuning, limiting robustness under multidimensional coupling and strong scatterers. We propose a range-domain sliding-window local tensorization that rearranges SLC data into localized range–azimuth–block-index tensors to better expose multi-mode correlations. On this representation, an energy-guided tensor CUR low-rank projector is embedded into an alternating-projection scheme that alternates complex-valued soft-thresholding for the sparse scene-plus-noise term and CUR-based projection for the structured RFI term. The cleaned SLC image is obtained by de-tensorizing the estimated RFI component and subtracting it from the input SLC. Experiments on semi-synthetic data, where controlled RFI is superimposed on real SLC scenes, and on real Sentinel-1 SLC data containing RFI demonstrate improved Pearson correlation coefficient (PCC) and perceptual image quality while preserving target signatures and scene textures, particularly under strong interference and strong coupling. The proposed approach provides a practical SLC-domain RFI mitigation tool for post-focusing SAR products without requiring explicit interference parameterization. Full article

With the continuous development of power systems, WAMS have become increasingly important for real-time system monitoring. As the core devices of WAMS, PMUs can provide synchronized, high-precision, and high-resolution measurements of power system states. However, in practical applications, PMUs deployed in different regions often operate at different sampling rates, resulting in multi-rate measurement data and posing challenges for data fusion. To address this issue, this paper proposes a multi-rate PMU data fusion method based on low-rank TT. Specifically, the proposed method first performs tensor-based modeling of multi-rate measurement data, embedding multidimensional correlations into a high-order tensor representation. Then, a data completion model is constructed through low-rank TT decomposition to effectively capture cross-timescale dependencies. Finally, an efficient numerical solution is developed to expand low-resolution measurements into high-resolution data, thereby achieving unified data fusion. Case studies on both simulated and real-world PMU measurement data demonstrate that the proposed approach outperforms traditional interpolation and matrix completion methods, achieving superior reconstruction accuracy and robustness. Full article

Elastic wavefield decomposition into quasi-compressional (qP) and quasi-shear-vertical (qSV) modes is essential for elastic imaging and inversion in VTI media, but becomes computationally expensive when polarization vectors vary strongly in space. I propose a tensor-train (TT) representation of mixed-domain decomposition projectors, constructed via TT-cross with a single user-specified tolerance and applied efficiently using FFT-based operations. A residual-orthogonal strategy extracts qSV from the residual wavefield after qP removal to suppress mode leakage. The method is implemented in Python/PyTorch with GPU acceleration. Numerical experiments on three 2D VTI models (a two-layer benchmark, a BP 2007 benchmark subset, and an Overthrust-based structurally complex model) demonstrate reconstruction errors of 0.094–0.89% for TT, compared to 1.67–6.44% for a conventional CUR low-rank approach (4–46× improvement), with consistently lower cross-talk and near-unity energy ratios. Time-domain receiver traces further confirm that TT yields smaller reconstruction residual spikes and reduced cross-mode leakage than CUR. Runtime tests show that CUR can be faster on smaller grids, whereas TT with GPU acceleration becomes competitive and can outperform CUR for larger models. The TT representation scales linearly with tensor $\mathcal{O}\left(d N_s r^2\right)$—enabling practical extension to higher-dimensional projector tensors where conven-tional methods become impractical. Full article

Projective Norms are a class of tensor norms that map from the input to the output spaces. These norms are useful for providing a measure of entanglement. Calculating the projective norms is an NP-hard problem, which creates challenges in computing due to the complexity of the exponentially growing parameter space for higher-order tensors. We develop a novel gradient descent algorithm to estimate the projective norm of higher-order tensors. The algorithm exhibits stable convergence to a minimum nuclear-rank decomposition of the given tensor in all our numerical experiments. We further extend the algorithm to symmetric tensors and to density matrices. We demonstrate the performance of our algorithm by computing the nuclear rank and the projective norm for both pure and mixed states and provide numerical evidence supporting these results. Full article

Accurate fault diagnosis of power transformers is critical for maintaining grid reliability, yet conventional dissolved gas analysis (DGA) methods face challenges in feature representation and high-dimensional data processing. This paper presents an intelligent diagnostic framework that synergistically integrates systematic feature engineering, tensor decomposition-based feature selection, and a sparrow search algorithm (SSA)-optimized multi-kernel support vector machine (MKSVM) for transformer fault classification. The proposed approach first expands the original five-dimensional gas concentration measurements to a twelve-dimensional feature space by incorporating domain-driven IEC 60599 ratio indicators and statistical aggregation descriptors, effectively capturing nonlinear interactions among gas components. Subsequently, a novel Tucker decomposition framework is developed to construct a three-way tensor encoding sample–feature–class relationships, where feature importance is quantified through both discriminative power and structural significance in low-rank representations, successfully reducing dimensionality from twelve to seven critical features while retaining 95% of discriminative information. The multi-kernel SVM architecture combines radial basis function, polynomial, and sigmoid kernels with optimized weights and hyperparameters configured through SSA’s hierarchical producer–scrounger search mechanism. Experimental validation on DGA samples across seven fault categories demonstrates that the proposed method achieves 98.33% classification accuracy, significantly outperforming existing methods, including kernel PCA-based approaches, deep learning models, and ensemble techniques. The framework establishes a reliable and accurate solution for transformer condition monitoring in power systems. Full article

Packing approaches enhance training efficiency by filling the padding space in each batch with shorter sequences, thereby reducing the total number of batches per epoch. This approach has proven effective in both pre-training and supervised fine-tuning of large language models (LLMs). However, most packing methods necessitate a packing-aware masking (PAM) mechanism to prevent cross-contamination between different text segments in the multi-head attention (MHA) layers. This masking ensures that the scaled dot-product attention operates only within segment boundaries. Despite its functional utility, PAM introduces significant implementation complexity and computational overhead during training. In this paper, we propose a novel method that eliminates the need for PAM during supervised fine-tuning with packing. Instead of masking, we introduce a learnable tensor derived from Low-Rank Adaptation (LoRA) with the query and value parameters of the attention mechanism. This tensor is trained to attenuate the subspace corresponding to cross-contamination, effectively replacing the function of PAM. Through component-wise decomposition of attention head outputs, we isolate the contamination component and demonstrate that it can be attenuated using the LoRA-derived tensor. Empirical evaluations on 7B-scale LLMs show that our method reduces training time and runtime overhead by completely removing the implementation associated with PAM. This enables more scalable and efficient supervised fine-tuning with packing, without compromising model integrity. Full article