Multilayer perceptrons (MLP), or fully connected artificial neural networks, are known for performing vector-matrix multiplications using learnable weight matrices; however, their practical application in many machine learning tasks, especially in computer vision, can be limited due to the high dimensionality of input-output pairs at each layer. To improve efficiency, convolutional operators have been utilized to facilitate weight sharing and local connections, yet they are constrained by limited receptive fields. In this paper, we introduce Multiscale Tensor Summation (MTS) Factorization, a novel neural network operator that implements tensor summation at multiple scales, where each tensor to be summed is obtained through Tucker-decomposition-like mode products. Unlike other tensor decomposition methods in the literature, MTS is not introduced as a network compression tool; instead, as a new backbone neural layer. MTS not only reduces the number of parameters required while enhancing the efficiency of weight optimization compared to traditional dense layers (i.e., unfactorized weight matrices in MLP layers), but it also demonstrates clear advantages over convolutional layers. The proof-of-concept experimental comparison of the proposed MTS networks with MLPs and Convolutional Neural Networks (CNNs) demonstrates their effectiveness across various tasks, such as classification, compression, and signal restoration. Additionally, when integrated with modern non-linear units such as the multi-head gate (MHG), also introduced in this study, the corresponding neural network, MTSNet, demonstrates a more favorable complexity-performance tradeoff compared to state-of-the-art transformers in various computer vision applications. The software implementation of the MTS layer and the corresponding MTS-based networks, MTSNets, is shared at https://github.com/mehmetyamac/MTSNet.
Implicit neural representation (INR) has emerged as a powerful paradigm for visual data representation. However, classical INR methods represent data in the original space mixed with different frequency components, and several feature encoding parameters (e.g., the frequency parameter $\omega$ or the rank $R$) need manual configurations. In this work, we propose a self-evolving cross-frequency INR using the Haar wavelet transform (termed CF-INR), which decouples data into four frequency components and employs INRs in the wavelet space. CF-INR allows the characterization of different frequency components separately, thus enabling higher accuracy for data representation. To more precisely characterize cross-frequency components, we propose a cross-frequency tensor decomposition paradigm for CF-INR with self-evolving parameters, which automatically updates the rank parameter $R$ and the frequency parameter $\omega$ for each frequency component through self-evolving optimization. This self-evolution paradigm eliminates the laborious manual tuning of these parameters, and learns a customized cross-frequency feature encoding configuration for each dataset. We evaluate CF-INR on a variety of visual data representation and recovery tasks, including image regression, inpainting, denoising, and cloud removal. Extensive experiments demonstrate that CF-INR outperforms state-of-the-art methods in each case.
The advancement of sensing technology has driven the widespread application of high-dimensional data. However, issues such as missing entries during acquisition and transmission negatively impact the accuracy of subsequent tasks. Tensor reconstruction aims to recover the underlying complete data from under-sampled observed data by exploring prior information in high-dimensional data. However, due to insufficient exploration, reconstruction methods still face challenges when sampling rate is extremely low. This work proposes a tensor reconstruction method integrating multiple priors to comprehensively exploit the inherent structure of the data. Specifically, the method combines learnable tensor decomposition to enforce low-rank constraints of the reconstructed data, a pre-trained convolutional neural network for smoothing and denoising, and block-matching and 3D filtering regularization to enhance the non-local similarity in the reconstructed data. An alternating direction method of the multipliers algorithm is designed to decompose the resulting optimization problem into three subproblems for efficient resolution. Extensive experiments on color images, hyperspectral images, and grayscale videos datasets demonstrate the superiority of our method in extreme cases as compared with state-of-the-art methods.
In this paper, we propose a new geometric approach for knowledge graph completion via low rank tensor approximation. We augment a pretrained and well-established Euclidean model based on a Tucker tensor decomposition with a novel hyperbolic interaction term. This correction enables more nuanced capturing of distributional properties in data better aligned with real-world knowledge graphs. By combining two geometries together, our approach improves expressivity of the resulting model achieving new state-of-the-art link prediction accuracy with a significantly lower number of parameters compared to the previous Euclidean and hyperbolic models.
The Khatri-Rao product is extensively used in array processing, tensor decomposition, and multi-way data analysis. Many applications require a least-squares (LS) Khatri-Rao factorization. In broadband sensor array problems, polynomial matrices effectively model frequency-dependent behaviors, necessitating extensions of conventional linear algebra techniques. This paper generalizes LS Khatri-Rao factorization from ordinary to polynomial matrices by applying it to the discrete Fourier transform (DFT) samples of polynomial matrices. Phase coherence across bin-wise Khatri-Rao factors is ensured via a phasesmoothing algorithm. The proposed method is validated through broadband angle-of-arrival (AoA) estimation for uniform planar arrays (UPAs), where the steering matrix is a polynomial matrix, which can be represented as a Khatri-Rao product between steering matrix in azimuth and elevation directions.
This study proposes a novel framework for long-term electricity demand prediction based solely on historical consumption data, without relying on external variables such as temperature or economic indicators. The method combines Non-negative Tensor Factorization (NTF) to extract low-dimensional temporal features from multi-way electricity usage data, with a Genetic Algorithm that optimizes the hyperparameters of time series models applied to the latent annual factors. We model the dataset as a third-order tensor spanning electric utilities, industrial sectors, and years, and apply canonical polyadic decomposition under non-negativity constraints. The annual component is forecasted using autoregressive models, with hyperparameter tuning guided by the prediction error or reconstruction accuracy on a validation set. Comparative experiments using real-world electricity data from Japan demonstrate that the proposed method achieves lower mean squared error than baseline approaches without tensor decomposition or evolutionary optimization. Moreover, we find that reducing the model's degrees of freedom via tensor decomposition improves generalization performance, and that initialization sensitivity in NTF can be mitigated through multiple runs or ensemble strategies. These findings suggest that the proposed framework offers an interpretable, flexible, and scalable approach to long-term electricity demand prediction and can be extended to other structured time series forecasting tasks.
Over recent years it has become well accepted that user interest is not static or immutable. There are a variety of contextual factors, such as time of day, the weather or the user's mood, that influence the current interests of the user. Modelling approaches need to take these factors into account if they want to succeed at finding the most relevant content to recommend given the situation. A popular method for context-aware recommendation is to encode context attributes as extra dimensions of the classic user-item interaction matrix, effectively turning it into a tensor, followed by applying the appropriate tensor decomposition methods to learn missing values. However, unlike with matrix factorization, where all decompositions are essentially a product of matrices, there exist many more options for decomposing tensors by combining vector, matrix and tensor products. We study the most successful decomposition methods that use weighted square loss and categorize them based on their tensor structure and regularization strategy. Additionally, we further extend the pool of methods by filling in the missing combinations. In this paper we provide an overview of the properties of the different decomposition methods, such as their complexity, scalability, and modelling capacity. These benefits are then contrasted with the performances achieved in offline experiments to gain more insight into which method to choose depending on a specific situation and constraints.
How can we accurately complete tensors by learning relationships of dimensions along each mode? Tensor completion, a widely studied problem, is to predict missing entries in incomplete tensors. Tensor decomposition methods, fundamental tensor analysis tools, have been actively developed to solve tensor completion tasks. However, standard tensor decomposition models have not been designed to learn relationships of dimensions along each mode, which limits to accurate tensor completion. Also, previously developed tensor decomposition models have required prior knowledge between relations within dimensions to model the relations, expensive to obtain. This paper proposes TGL (Tensor Decomposition Learning Global and Local Structures) to accurately predict missing entries in tensors. TGL reconstructs a tensor with factor matrices which learn local structures with GNN without prior knowledges. Extensive experiments are conducted to evaluate TGL with baselines and datasets.
Implicit Neural Representation (INR) has emerged as a powerful tool for encoding discrete signals into continuous, differentiable functions using neural networks. However, these models often have an unfortunate reliance on monolithic architectures to represent high-dimensional data, leading to prohibitive computational costs as dimensionality grows. We propose F-INR, a framework that reformulates INR learning through functional tensor decomposition, breaking down high-dimensional tasks into lightweight, axis-specific sub-networks. Each sub-network learns a low-dimensional data component (e.g., spatial or temporal). Then, we combine these components via tensor operations, reducing forward pass complexity while improving accuracy through specialized learning. F-INR is modular and, therefore, architecture-agnostic, compatible with MLPs, SIREN, WIRE, or other state-of-the-art INR architecture. It is also decomposition-agnostic, supporting CP, TT, and Tucker modes with user-defined rank for speed-accuracy control. In our experiments, F-INR trains $100\times$ faster than existing approaches on video tasks while achieving higher fidelity (+3.4 dB PSNR). Similar gains hold for image compression, physics simulations, and 3D geometry reconstruction. Through this, F-INR offers a new scalable, flexible solution for high-dimensional signal modeling.
Transformer-based large pre-trained models have shown remarkable generalization ability, and various parameter-efficient fine-tuning (PEFT) methods have been proposed to customize these models on downstream tasks with minimal computational and memory budgets. Previous PEFT methods are primarily designed from a tensor-decomposition perspective that tries to effectively tune the linear transformation by finding the smallest subset of parameters to train. Our study adopts an orthogonal view by representing the attention operation as a graph convolution and formulating the multi-head attention maps as a convolutional filter subspace, with each attention map as a subspace element. In this paper, we propose to tune the large pre-trained transformers by learning a small set of combination coefficients that construct a more expressive filter subspace from the original multi-head attention maps. We show analytically and experimentally that the tuned filter subspace can effectively expand the feature space of the multi-head attention and further enhance the capacity of transformers. We further stabilize the fine-tuning with a residual parameterization of the tunable subspace coefficients, and enhance the generalization with a regularization design by directly applying dropout on the tunable coefficient during training. The tunable coefficients take a tiny number of parameters and can be combined with previous PEFT methods in a plug-and-play manner. Extensive experiments show that our approach achieves superior performances than PEFT baselines with neglectable additional parameters.
Low rank tensor representation (LRTR) methods are very useful for hyperspectral anomaly detection (HAD). To overcome the limitations that they often overlook spectral anomaly and rely on large-scale matrix singular value decomposition, we first apply non-negative matrix factorization (NMF) to alleviate spectral dimensionality redundancy and extract spectral anomaly and then employ LRTR to extract spatial anomaly while mitigating spatial redundancy, yielding a highly efffcient layered tensor decomposition (LTD) framework for HAD. An iterative algorithm based on proximal alternating minimization is developed to solve the proposed LTD model, with convergence guarantees provided. Moreover, we introduce a rank reduction strategy with validation mechanism that adaptively reduces data size while preventing excessive reduction. Theoretically, we rigorously establish the equivalence between the tensor tubal rank and tensor group sparsity regularization (TGSR) and, under mild conditions, demonstrate that the relaxed formulation of TGSR shares the same global minimizers and optimal values as its original counterpart. Experimental results on the Airport-Beach-Urban and MVTec datasets demonstrate that our approach outperforms state-of-the-art methods in the HAD task.
We present novel algorithmic techniques to efficiently verify the Kruskal rank of matrices that arise in sparse linear regression, tensor decomposition, and latent variable models. Our unified framework combines randomized hashing techniques with dynamic programming strategies, and is applicable in various settings, including binary fields, general finite fields, and integer matrices. In particular, our algorithms achieve a runtime of $\mathcal{O}\left(dk \cdot \left(nM\right)^{\lceil k / 2 \rceil}\right)$ while ensuring high-probability correctness. Our contributions include: A unified framework for verifying Kruskal rank across different algebraic settings; Rigorous runtime and high-probability guarantees that nearly match known lower bounds; Practical implications for identifiability in tensor decompositions and deep learning, particularly for the estimation of noise transition matrices.
Large Language Models (LLMs) have demonstrated remarkable capabilities across various tasks due to large training datasets and powerful transformer architecture. However, the reliability of responses from LLMs remains a question. Uncertainty quantification (UQ) of LLMs is crucial for ensuring their reliability, especially in areas such as healthcare, finance, and decision-making. Existing UQ methods primarily focus on semantic similarity, overlooking the deeper knowledge dimensions embedded in responses. We introduce a multi-dimensional UQ framework that integrates semantic and knowledge-aware similarity analysis. By generating multiple responses and leveraging auxiliary LLMs to extract implicit knowledge, we construct separate similarity matrices and apply tensor decomposition to derive a comprehensive uncertainty representation. This approach disentangles overlapping information from both semantic and knowledge dimensions, capturing both semantic variations and factual consistency, leading to more accurate UQ. Our empirical evaluations demonstrate that our method outperforms existing techniques in identifying uncertain responses, offering a more robust framework for enhancing LLM reliability in high-stakes applications.
Tensor decomposition has emerged as a powerful framework for feature extraction in multi-modal biomedical data. In this review, we present a comprehensive analysis of tensor decomposition methods such as Tucker, CANDECOMP/PARAFAC, spiked tensor decomposition, etc. and their diverse applications across biomedical domains such as imaging, multi-omics, and spatial transcriptomics. To systematically investigate the literature, we applied a topic modeling-based approach that identifies and groups distinct thematic sub-areas in biomedicine where tensor decomposition has been used, thereby revealing key trends and research directions. We evaluated challenges related to the scalability of latent spaces along with obtaining the optimal rank of the tensor, which often hinder the extraction of meaningful features from increasingly large and complex datasets. Additionally, we discuss recent advances in quantum algorithms for tensor decomposition, exploring how quantum computing can be leveraged to address these challenges. Our study includes a preliminary resource estimation analysis for quantum computing platforms and examines the feasibility of implementing quantum-enhanced tensor decomposition methods on near-term quantum devices. Collectively, this review not only synthesizes current applications and challenges of tensor decomposition in biomedical analyses but also outlines promising quantum computing strategies to enhance its impact on deriving actionable insights from complex biomedical data.
We study tensor completion (TC) through the lens of low-rank tensor decomposition (TD). Many TD algorithms use fast alternating minimization methods, which solve highly structured linear regression problems at each step (e.g., for CP, Tucker, and tensor-train decompositions). However, such algebraic structure is lost in TC regression problems, making direct extensions unclear. To address this, we propose a lifting approach that approximately solves TC regression problems using structured TD regression algorithms as blackbox subroutines, enabling sublinear-time methods. We theoretically analyze the convergence rate of our approximate Richardson iteration based algorithm, and we demonstrate on real-world tensors that its running time can be 100x faster than direct methods for CP completion.
Low-rank tensor estimation offers a powerful approach to addressing high-dimensional data challenges and can substantially improve solutions to ill-posed inverse problems, such as image reconstruction under noisy or undersampled conditions. Meanwhile, tensor decomposition has gained prominence in federated learning (FL) due to its effectiveness in exploiting latent space structure and its capacity to enhance communication efficiency. In this paper, we present a federated image reconstruction method that applies Tucker decomposition, incorporating joint factorization and randomized sketching to manage large-scale, multimodal data. Our approach avoids reconstructing full-size tensors and supports heterogeneous ranks, allowing clients to select personalized decomposition ranks based on prior knowledge or communication capacity. Numerical results demonstrate that our method achieves superior reconstruction quality and communication compression compared to existing approaches, thereby highlighting its potential for multimodal inverse problems in the FL setting.
Recently, coupled tensor decomposition has been widely used in data fusion of a hyperspectral image (HSI) and a multispectral image (MSI) for hyperspectral super-resolution (HSR). However, exsiting works often ignore the inherent non-negative (NN) property of the image data, or impose the NN constraint via hard-thresholding which may interfere with the optimization procedure and cause the method to be sub-optimal. As such, we propose a novel NN coupled canonical polyadic decomposition (NN-C-CPD) algorithm, which makes use of the parametric method and nonlinear least squares (NLS) framework to impose the NN constraint into the C-CPD computation. More exactly, the NN constraint is converted into the squared relationship between the NN entries of the factor matrices and a set of latent parameters. Based on the chain rule for deriving the derivatives, the key entities such as gradient and Jacobian with regards to the latent parameters can be derived, thus the NN constraint is naturally integrated without interfering with the optimization procedure. Experimental results are provided to demonstrate the performance of the proposed NN-C-CPD algorithm in HSR applications.
In recent years, the prediction of multidimensional time series data has become increasingly important due to its wide-ranging applications. Tensor-based prediction methods have gained attention for their ability to preserve the inherent structure of such data. However, existing approaches, such as tensor autoregression and tensor decomposition, often have consistently failed to provide clear assertions regarding the number of samples that can be exactly predicted. While matrix-based methods using nuclear norms address this limitation, their reliance on matrices limits accuracy and increases computational costs when handling multidimensional data. To overcome these challenges, we reformulate multidimensional time series prediction as a deterministic tensor completion problem and propose a novel theoretical framework. Specifically, we develop a deterministic tensor completion theory and introduce the Temporal Convolutional Tensor Nuclear Norm (TCTNN) model. By convolving the multidimensional time series along the temporal dimension and applying the tensor nuclear norm, our approach identifies the maximum forecast horizon for exact predictions. Additionally, TCTNN achieves superior performance in prediction accuracy and computational efficiency compared to existing methods across diverse real-world datasets, including climate temperature, network flow, and traffic ride data. Our implementation is publicly available at https://github.com/HaoShu2000/TCTNN.
Resource-constrained edge deployments demand AI solutions that balance high performance with stringent compute, memory, and energy limitations. In this survey, we present a comprehensive overview of the primary strategies for accelerating deep learning models under such constraints. First, we examine model compression techniques-pruning, quantization, tensor decomposition, and knowledge distillation-that streamline large models into smaller, faster, and more efficient variants. Next, we explore Neural Architecture Search (NAS), a class of automated methods that discover architectures inherently optimized for particular tasks and hardware budgets. We then discuss compiler and deployment frameworks, such as TVM, TensorRT, and OpenVINO, which provide hardware-tailored optimizations at inference time. By integrating these three pillars into unified pipelines, practitioners can achieve multi-objective goals, including latency reduction, memory savings, and energy efficiency-all while maintaining competitive accuracy. We also highlight emerging frontiers in hierarchical NAS, neurosymbolic approaches, and advanced distillation tailored to large language models, underscoring open challenges like pre-training pruning for massive networks. Our survey offers practical insights, identifies current research gaps, and outlines promising directions for building scalable, platform-independent frameworks to accelerate deep learning models at the edge.
Parameter-Efficient Fine-Tuning (PEFT) of text-to-image models has become an increasingly popular technique with many applications. Among the various PEFT methods, Low-Rank Adaptation (LoRA) and its variants have gained significant attention due to their effectiveness, enabling users to fine-tune models with limited computational resources. However, the approximation gap between the low-rank assumption and desired fine-tuning weights prevents the simultaneous acquisition of ultra-parameter-efficiency and better performance. To reduce this gap and further improve the power of LoRA, we propose a new PEFT method that combines two classes of adaptations, namely, transform and residual adaptations. In specific, we first apply a full-rank and dense transform to the pre-trained weight. This learnable transform is expected to align the pre-trained weight as closely as possible to the desired weight, thereby reducing the rank of the residual weight. Then, the residual part can be effectively approximated by more compact and parameter-efficient structures, with a smaller approximation error. To achieve ultra-parameter-efficiency in practice, we design highly flexible and effective tensor decompositions for both the transform and residual adaptations. Additionally, popular PEFT methods such as DoRA can be summarized under this transform plus residual adaptation scheme. Experiments are conducted on fine-tuning Stable Diffusion models in subject-driven and controllable generation. The results manifest that our method can achieve better performances and parameter efficiency compared to LoRA and several baselines.