Canonical Polyadic (CP) tensor decomposition is a fundamental technique for analyzing high-dimensional tensor data. While the Alternating Least Squares (ALS) algorithm is widely used for computing CP decomposition due to its simplicity and empirical success, its theoretical foundation, particularly regarding statistical optimality and convergence behavior, remain underdeveloped, especially in noisy, non-orthogonal, and higher-rank settings. In this work, we revisit CP tensor decomposition from a statistical perspective and provide a comprehensive theoretical analysis of ALS under a signal-plus-noise model. We establish non-asymptotic, minimax-optimal error bounds for tensors of general order, dimensions, and rank, assuming suitable initialization. To enable such initialization, we propose Tucker-based Approximation with Simultaneous Diagonalization (TASD), a robust method that improves stability and accuracy in noisy regimes. Combined with ALS, TASD yields a statistically consistent estimator. We further analyze the convergence dynamics of ALS, identifying a two-phase pattern-initial quadratic convergence followed by linear refinement. We further show that in the rank-one setting, ALS with an appropriately chosen initialization attains optimal error within just one or two iterations.
Large quantities of social activity data, such as weekly web search volumes and the number of new infections with infectious diseases, reflect peoples' interests and activities. It is important to discover temporal patterns from such data and to forecast future activities accurately. However, modeling and forecasting social activity data streams is difficult because they are high-dimensional and composed of multiple time-varying dynamics such as trends, seasonality, and interest diffusion. In this paper, we propose D-Tracker, a method for continuously capturing time-varying temporal patterns within social activity tensor data streams and forecasting future activities. Our proposed method has the following properties: (a) Interpretable: it incorporates the partial differential equation into a tensor decomposition framework and captures time-varying temporal patterns such as trends, seasonality, and interest diffusion between locations in an interpretable manner; (b) Automatic: it has no hyperparameters and continuously models tensor data streams fully automatically; (c) Scalable: the computation time of D-Tracker is independent of the time series length. Experiments using web search volume data obtained from GoogleTrends, and COVID-19 infection data obtained from COVID-19 Open Data Repository show that our method can achieve higher forecasting accuracy in less computation time than existing methods while extracting the interest diffusion between locations. Our source code and datasets are available at {https://github.com/Higashiguchi-Shingo/D-Tracker.
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.
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.
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.
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.
Irreducible Cartesian tensors (ICTs) play a crucial role in the design of equivariant graph neural networks, as well as in theoretical chemistry and chemical physics. Meanwhile, the design space of available linear operations on tensors that preserve symmetry presents a significant challenge. The ICT decomposition and a basis of this equivariant space are difficult to obtain for high-rank tensors. After decades of research, Bonvicini (2024) recently achieves an explicit ICT decomposition for $n=5$ with factorial time/space complexity. In this work we, for the first time, obtains decomposition matrices for ICTs up to rank $n=9$ with reduced and affordable complexity, by constructing what we call path matrices. The path matrices are obtained via performing chain-like contractions with Clebsch-Gordan matrices following the parentage scheme. We prove and leverage that the concatenation of path matrices is an orthonormal change-of-basis matrix between the Cartesian tensor product space and the spherical direct sum spaces. Furthermore, we identify a complete orthogonal basis for the equivariant space, rather than a spanning set (Pearce-Crump, 2023), through this path matrices technique. To the best of our knowledge, this is also the first analytic, rather than numerical, method for theoretically obtaining arbitrary rank orthogonal ICT decomposition matrices and orthogonal equivariant bases. We further extend our result to the arbitrary tensor product and direct sum spaces, enabling free design between different spaces while keeping symmetry. The Python code is available at https://github.com/ShihaoShao-GH/ICT-decomposition-and-equivariant-bases, where the $n=6,\dots,9$ ICT decomposition matrices are obtained in 1s, 3s, 11s, and 4m32s on 28-cores Intel(R) Xeon(R) Gold 6330 CPU @ 2.00GHz, respectively.
Increased training parameters have enabled large pre-trained models to excel in various downstream tasks. Nevertheless, the extensive computational requirements associated with these models hinder their widespread adoption within the community. We focus on Knowledge Distillation (KD), where a compact student model is trained to mimic a larger teacher model, facilitating the transfer of knowledge of large models. In contrast to much of the previous work, we scale up the parameters of the student model during training, to benefit from overparameterization without increasing the inference latency. In particular, we propose a tensor decomposition strategy that effectively over-parameterizes the relatively small student model through an efficient and nearly lossless decomposition of its parameter matrices into higher-dimensional tensors. To ensure efficiency, we further introduce a tensor constraint loss to align the high-dimensional tensors between the student and teacher models. Comprehensive experiments validate the significant performance enhancement by our approach in various KD tasks, covering computer vision and natural language processing areas. Our code is available at https://github.com/intell-sci-comput/OPDF.
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.
This work presents a low-rank tensor model for multi-dimensional Markov chains. A common approach to simplify the dynamical behavior of a Markov chain is to impose low-rankness on the transition probability matrix. Inspired by the success of these matrix techniques, we present low-rank tensors for representing transition probabilities on multi-dimensional state spaces. Through tensor decomposition, we provide a connection between our method and classical probabilistic models. Moreover, our proposed model yields a parsimonious representation with fewer parameters than matrix-based approaches. Unlike these methods, which impose low-rankness uniformly across all states, our tensor method accounts for the multi-dimensionality of the state space. We also propose an optimization-based approach to estimate a Markov model as a low-rank tensor. Our optimization problem can be solved by the alternating direction method of multipliers (ADMM), which enjoys convergence to a stationary solution. We empirically demonstrate that our tensor model estimates Markov chains more efficiently than conventional techniques, requiring both fewer samples and parameters. We perform numerical simulations for both a synthetic low-rank Markov chain and a real-world example with New York City taxi data, showcasing the advantages of multi-dimensionality for modeling state spaces.
Convolutional neural networks (CNNs) have made significant advances in computer vision tasks, yet their high inference times and latency often limit real-world applicability. While model compression techniques have gained popularity as solutions, they often overlook the critical balance between low latency and uncompromised accuracy. By harnessing quantum-inspired pruning, tensor decomposition, and annealing-based matrix factorization - three quantum-inspired concepts - we introduce QIANets: a novel approach of redesigning the traditional GoogLeNet, DenseNet, and ResNet-18 model architectures to process more parameters and computations whilst maintaining low inference times. Despite experimental limitations, the method was tested and evaluated, demonstrating reductions in inference times, along with effective accuracy preservations.
Anomaly detection (AD) is increasingly recognized as a key component for ensuring the resilience of future communication systems. While deep learning has shown state-of-the-art AD performance, its application in critical systems is hindered by concerns regarding training data efficiency, domain adaptation and interpretability. This work considers AD in network flows using incomplete measurements, leveraging a robust tensor decomposition approach and deep unrolling techniques to address these challenges. We first propose a novel block-successive convex approximation algorithm based on a regularized model-fitting objective where the normal flows are modeled as low-rank tensors and anomalies as sparse. An augmentation of the objective is introduced to decrease the computational cost. We apply deep unrolling to derive a novel deep network architecture based on our proposed algorithm, treating the regularization parameters as learnable weights. Inspired by Bayesian approaches, we extend the model architecture to perform online adaptation to per-flow and per-time-step statistics, improving AD performance while maintaining a low parameter count and preserving the problem's permutation equivariances. To optimize the deep network weights for detection performance, we employ a homotopy optimization approach based on an efficient approximation of the area under the receiver operating characteristic curve. Extensive experiments on synthetic and real-world data demonstrate that our proposed deep network architecture exhibits a high training data efficiency, outperforms reference methods, and adapts seamlessly to varying network topologies.
Physics-Informed Neural Networks (PINNs) have shown continuous and increasing promise in approximating partial differential equations (PDEs), although they remain constrained by the curse of dimensionality. In this paper, we propose a generalized PINN version of the classical variable separable method. To do this, we first show that, using the universal approximation theorem, a multivariate function can be approximated by the outer product of neural networks, whose inputs are separated variables. We leverage tensor decomposition forms to separate the variables in a PINN setting. By employing Canonic Polyadic (CP), Tensor-Train (TT), and Tucker decomposition forms within the PINN framework, we create robust architectures for learning multivariate functions from separate neural networks connected by outer products. Our methodology significantly enhances the performance of PINNs, as evidenced by improved results on complex high-dimensional PDEs, including the 3d Helmholtz and 5d Poisson equations, among others. This research underscores the potential of tensor decomposition-based variably separated PINNs to surpass the state-of-the-art, offering a compelling solution to the dimensionality challenge in PDE approximation.
Recently, introducing Tensor Decomposition (TD) methods into unsupervised feature selection (UFS) has been a rising research point. A tensor structure is beneficial for mining the relations between different modes and helps relieve the computation burden. However, while existing methods exploit TD to minimize the reconstruction error of a data tensor, they don't fully utilize the interpretable and discriminative information in the factor matrices. Moreover, most methods require domain knowledge to perform feature selection. To solve the above problems, we develop two Sparse Tensor Principal Component Analysis (STPCA) models that utilize the projection directions in the factor matrices to perform UFS. The first model extends Tucker Decomposition to a multiview sparse regression form and is transformed into several alternatively solved convex subproblems. The second model formulates a sparse version of the family of Tensor Singular Value Decomposition (T-SVDs) and is transformed into individual convex subproblems. For both models, we prove the optimal solution of each subproblem falls onto the Hermitian Positive Semidefinite Cone (HPSD). Accordingly, we design two fast algorithms based on HPSD projection and prove their convergence. According to the experimental results on two original synthetic datasets (Orbit and Array Signal) and five real-world datasets, the two proposed methods are suitable for handling different data tensor scenarios and outperform the state-of-the-art UFS methods.
Infrared small target detection is crucial for the efficacy of infrared search and tracking systems. Current tensor decomposition methods emphasize representing small targets with sparsity but struggle to separate targets from complex backgrounds due to insufficient use of intrinsic directional information and reduced target visibility during decomposition. To address these challenges, this study introduces a Sparse Differential Directionality prior (SDD) framework. SDD leverages the distinct directional characteristics of targets to differentiate them from the background, applying mixed sparse constraints on the differential directional images and continuity difference matrix of the temporal component, both derived from Tucker decomposition. We further enhance target detectability with a saliency coherence strategy that intensifies target contrast against the background during hierarchical decomposition. A Proximal Alternating Minimization-based (PAM) algorithm efficiently solves our proposed model. Experimental results on several real-world datasets validate our method's effectiveness, outperforming ten state-of-the-art methods in target detection and clutter suppression. Our code is available at https://github.com/GrokCV/SDD.
In recent years, there has been growing interest in jointly analyzing a foreground dataset, representing an experimental group, and a background dataset, representing a control group. The goal of such contrastive investigations is to identify salient features in the experimental group relative to the control. Independent component analysis (ICA) is a powerful tool for learning independent patterns in a dataset. We generalize it to contrastive ICA (cICA). For this purpose, we devise a new linear algebra based tensor decomposition algorithm, which is more expressive but just as efficient and identifiable as other linear algebra based algorithms. We establish the identifiability of cICA and demonstrate its performance in finding patterns and visualizing data, using synthetic, semi-synthetic, and real-world datasets, comparing the approach to existing methods.
Second-order Recurrent Neural Networks (2RNNs) extend RNNs by leveraging second-order interactions for sequence modelling. These models are provably more expressive than their first-order counterparts and have connections to well-studied models from formal language theory. However, their large parameter tensor makes computations intractable. To circumvent this issue, one approach known as MIRNN consists in limiting the type of interactions used by the model. Another is to leverage tensor decomposition to diminish the parameter count. In this work, we study the model resulting from parameterizing 2RNNs using the CP decomposition, which we call CPRNN. Intuitively, the rank of the decomposition should reduce expressivity. We analyze how rank and hidden size affect model capacity and show the relationships between RNNs, 2RNNs, MIRNNs, and CPRNNs based on these parameters. We support these results empirically with experiments on the Penn Treebank dataset which demonstrate that, with a fixed parameter budget, CPRNNs outperforms RNNs, 2RNNs, and MIRNNs with the right choice of rank and hidden size.
Deep Click-Through Rate (CTR) prediction models play an important role in modern industrial recommendation scenarios. However, high memory overhead and computational costs limit their deployment in resource-constrained environments. Low-rank approximation is an effective method for computer vision and natural language processing models, but its application in compressing CTR prediction models has been less explored. Due to the limited memory and computing resources, compression of CTR prediction models often confronts three fundamental challenges, i.e., (1). How to reduce the model sizes to adapt to edge devices? (2). How to speed up CTR prediction model inference? (3). How to retain the capabilities of original models after compression? Previous low-rank compression research mostly uses tensor decomposition, which can achieve a high parameter compression ratio, but brings in AUC degradation and additional computing overhead. To address these challenges, we propose a unified low-rank decomposition framework for compressing CTR prediction models. We find that even with the most classic matrix decomposition SVD method, our framework can achieve better performance than the original model. To further improve the effectiveness of our framework, we locally compress the output features instead of compressing the model weights. Our unified low-rank compression framework can be applied to embedding tables and MLP layers in various CTR prediction models. Extensive experiments on two academic datasets and one real industrial benchmark demonstrate that, with 3-5x model size reduction, our compressed models can achieve both faster inference and higher AUC than the uncompressed original models. Our code is at https://github.com/yuhao318/Atomic_Feature_Mimicking.
Adapting pre-trained foundation models for various downstream tasks has been prevalent in artificial intelligence. Due to the vast number of tasks and high costs, adjusting all parameters becomes unfeasible. To mitigate this, several fine-tuning techniques have been developed to update the pre-trained model weights in a more resource-efficient manner, such as through low-rank adjustments. Yet, almost all of these methods focus on linear weights, neglecting the intricacies of parameter spaces in higher dimensions like 4D. Alternatively, some methods can be adapted for high-dimensional parameter space by compressing changes in the original space into two dimensions and then employing low-rank matrix decomposition. However, these approaches destructs the structural integrity of the involved high-dimensional spaces. To tackle the diversity of dimensional spaces across different foundation models and provide a more precise representation of the changes within these spaces, this paper introduces a generalized parameter-efficient fine-tuning framework, FLoRA, designed for various dimensional parameter space. Specifically, utilizing Tucker decomposition, FLoRA asserts that changes in each dimensional parameter space are based on a low-rank core space which maintains the consistent topological structure with the original space. It then models the changes through this core space alongside corresponding weights to reconstruct alterations in the original space. FLoRA effectively preserves the structural integrity of the change of original N-dimensional parameter space, meanwhile decomposes it via low-rank tensor decomposition. Extensive experiments on computer vision, natural language processing and multi-modal tasks validate FLoRA's effectiveness. Codes are available at https://github.com/SJTU-DeepVisionLab/FLoRA.
Key-value~(KV) caching is an important technique to accelerate the inference of large language models~(LLMs), but incurs significant memory overhead. To compress the size of KV cache, existing methods often compromise precision or require extra data for calibration, limiting their practicality in LLM deployment. In this paper, we introduce \textbf{DecoQuant}, a novel data-free low-bit quantization technique based on tensor decomposition methods, to effectively compress KV cache. Our core idea is to adjust the outlier distribution of the original matrix by performing tensor decomposition, so that the quantization difficulties are migrated from the matrix to decomposed local tensors. Specially, we find that outliers mainly concentrate on small local tensors, while large tensors tend to have a narrower value range. Based on this finding, we propose to apply low-bit quantization to the large tensor, while maintaining high-precision representation for the small tensor. Furthermore, we utilize the proposed quantization method to compress the KV cache of LLMs to accelerate the inference and develop an efficient dequantization kernel tailored specifically for DecoQuant. Through extensive experiments, DecoQuant demonstrates remarkable efficiency gains, showcasing up to a $\sim$75\% reduction in memory footprint while maintaining comparable generation quality.