Block Term Decomposition Research Articles

Block-term tensor decomposition in array signal processing

The ability of tensor decomposition (TD)-based methods to recover latent information, including channel parameters and transmitted symbols in the array signal processing (ASP) context, in a deterministic manner and with little or no training overhead, fits well with the data efficiency needs of future-generation multi-user massive multiple-input multiple-output systems. However, such methods have mostly relied on a few classical TD models, notably the canonical polyadic decomposition (CPD), which can be quite restrictive in realistic scenarios. In this paper, a generalized CPD model, known as block-term decomposition (BTD), is re-visited in the context of ASP, and shown to be the natural choice in sensor arrays of increased dimensionality that also involve channel multipath. Semi-blind joint channel estimation/data detection (JCD) is addressed in this context via efficient algorithms that wed existing JCD schemes with BTD approximation. Robust to sensor failures and recursive versions are also developed. The special yet important case of the uniform rectangular array (URA) configuration is adopted to illustrate the ideas and results. The signal detection performance of the BTD-inspired semi-blind JCD schemes is evaluated under various conditions with the aid of simulations and seen to be favorably compared with that of the training-only-based solution.

Tensor decomposition meets blind source separation

In this paper, we investigate the problem of blind source separation (BSS) through the lens of tensor decomposition (TD). Two fundamental connections between TD and BSS are established, forming the basis for two novel tensor-based BSS methods, namely TenSOFO and TCBSS. The former is designed for a joint individual differences in scaling (INDSCAL) decomposition, addressing instantaneous (linear) BSS tasks; while the latter efficiently performs a constrained block term decomposition (BTD), aligning with the design of convolutive BSS. Leveraging the benefits of the alternating direction method of multipliers and the strengths of tensor representations, both TenSOFO and TCBSS prove to be effective in BSS. Our experimental results demonstrate the effectiveness of these two proposed methods in addressing both TD and BSS tasks, particularly when compared to state-of-the-art algorithms.

Probabilistic Block Term Decomposition for the Modelling of Higher-order Arrays

Probabilistic Block Term Decomposition for the Modelling of Higher-order Arrays

Multimodal Fusion With Block Term Decomposition for Asynchronous Federated Learning

Multimodal Fusion With Block Term Decomposition for Asynchronous Federated Learning

IMep: Device-Free Multiplayer Step Counting With WiFi Signals

Currently, most of the mature WiFi-based step applications can only count the steps of a single individual, and their methods cannot capture amplitude information about each person’s overall actions in multiuser scenes. It is still a challenging task to build a multiplayer step counting system in a device-free manner. In this paper, we present IMep, a novel device-free system based on WiFi that can obtain the amplitude information of each person’s overall step actions and count the steps of multiple people simultaneously. Our main strategy is to establish a Multiplayer Stepping Amplitude Relation Model (MSARM) and design a Multiplayer Amplitude Decomposition Algorithm (MADA) that uses Block Term Decomposition (BTD). Moreover, we put forward a new Moving Energy Method (MEM) that captures each person’s step number more clearly and accurately. The experimental results show that, IMep can function successfully in an environment of up to 7 people. The accuracies in three different room settings are 95.57%, 94.66%, and 89.94%, respectively, suggesting that IMep is effective in multi-person scenarios.

Online rank-revealing block-term tensor decomposition

The so-called block-term decomposition (BTD) tensor model, especially in its rank-(Lr,Lr,1) version, has been recently receiving increasing attention due to its enhanced ability to represent systems and signals that are composed of block components of rank higher than one, a scenario encountered in numerous and diverse applications. Its uniqueness and approximation have thus been thoroughly studied. The challenging problem of estimating the BTD model structure, namely the number of block terms (rank) and their individual (block) ranks, is of crucial importance in practice and has only recently started to attract significant attention. In data-streaming scenarios and/or big data applications, where the tensor size grows in time or the processing can only be done incrementally, it is essential to be able to perform model selection and computation in a recursive (online/incremental) manner. In this paper, a novel approach to rank-(Lr,Lr,1) BTD model selection and tracking is proposed, based on the idea of imposing column sparsity jointly on the factors and estimating the ranks as the numbers of factor columns of non-negligible magnitude. In this vein, and using a new rank-revealing batch BTD approximation algorithm as the starting point, an online method of the alternating reweighted least squares (RLS) type is developed and shown to be computationally efficient and fast converging, also allowing the model ranks to change in time. Its time and memory efficiency are evaluated and favorably compared with those of the batch approach and a recently reported online BTD scheme that is based on a-priori knowledge of the ranks. Simulation results with both synthetic and real (video) data are reported, that demonstrate the effectiveness of the proposed scheme in both selecting and tracking the correct BTD model.

Dynamic Community Detection for Brain Functional Networks During Music Listening With Block Component Analysis

The human brain can be described as a complex network of functional connections between distinct regions, referred to as the brain functional network. Recent studies show that the functional network is a dynamic process and its community structure evolves with time during continuous task performance. Consequently, it is important for the understanding of the human brain to develop dynamic community detection techniques for such time-varying functional networks. Here, we propose a temporal clustering framework based on a set of network generative models and surprisingly it can be linked to Block Component Analysis to detect and track the latent community structure in dynamic functional networks. Specifically, the temporal dynamic networks are represented within a unified three-way tensor framework for simultaneously capturing multiple types of relationships between a set of entities. The multi-linear rank- (Lr, Lr, 1) block term decomposition (BTD) is adopted to fit the network generative model to directly recover underlying community structures with the specific evolution of time from the temporal networks. We apply the proposed method to the study of the reorganization of the dynamic brain networks from electroencephalography (EEG) data recorded during free music listening. We derive several network structures ( Lr communities in each component) with specific temporal patterns (described by BTD components) significantly modulated by musical features, involving subnetworks of frontoparietal, default mode, and sensory-motor networks. The results show that the brain functional network structures are dynamically reorganized and the derived community structures are temporally modulated by the music features. The proposed generative modeling approach can be an effective tool for describing community structures in brain networks that go beyond static methods and detecting the dynamic reconfiguration of modular connectivity elicited by continuously naturalistic tasks.

Hyperspectral Image Mixed Noise Removal via Nonlinear Transform-Based Block-Term Tensor Decomposition

Recently, block-term decomposition with rank-( <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">L_r</i> , <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">L_r</i> ,1) (termed as LL1 decomposition), which is physically inspired by linear spectral unmixing, has received increasing attention in hyperspectral images (HSIs) denoising. However, due to the intrinsic nonlinear structure of real-world HSIs, the low-rankness of HSIs is usually implicit. Moreover, the essential uniqueness guarantee is usually violated with the low-rank assumption of the abundance maps unsupported in real scenarios, which hampers the successful deployment of LL1 decomposition. Inspired by the nonlinear spectral unmixing, we propose a nonlinear learnable transform-based LL1 decomposition (NT-LL1) for characterizing the implicit low-rank structure of real-world HSIs. More concretely, the nonlinear learnable transform in NT-LL1 decomposition is a composed transform consisting of a linear semi-orthogonal transform and a component-wise nonlinear transform, which collaboratively enhances the low-rankness of the abundance maps. Empowering with the NT-LL1 decomposition, we propose an NT-LL1 decomposition-based model for HSIs denoising. To tackle the resulting model, we develop an efficient proximal alternating minimization-based algorithm with a convergence guarantee. Extensive experimental results including simulated and real data collectively verify the superiority of the proposed method as compared with the competing methods.

MIMO transfer function decoupling by Loewner tensorization⋆

Dynamic systems often have an underlying latent structure that is desirable to uncover during a modeling step or to exploit in applications. Here, we consider MIMO systems that admit a diagonal (or dyadic) form in which the system dynamics are contained in a structure of internal SISO systems surrounded by static mixing matrices that relate the internal dynamics to the inputs and outputs. The fact that linear dynamic systems are described by rational transfer functions naturally leads to a Loewner-based tensor method in which the latent structure is identified by means of a block-term decomposition of a fourth-order tensor built from Loewner matrices constructed from the MIMO input-output data. Numerical experiments illustrate how the proposed decoupling method improves the state-of-the-art approach in terms of guaranteed conditions for the unique retrieval of the structure, and exhibit an improved ability to reject noise.

Enhanced Solutions for the Block-Term Decomposition in Rank-$(L_{r},L_{r},1)$ Terms

Enhanced Solutions for the Block-Term Decomposition in Rank-$(L_{r},L_{r},1)$ Terms

Deconvolution of the Functional Ultrasound Response in the Mouse Visual Pathway Using Block-Term Decomposition

Functional ultrasound (fUS) indirectly measures brain activity by detecting changes in cerebral blood volume following neural activation. Conventional approaches model such functional neuroimaging data as the convolution between an impulse response, known as the hemodynamic response function (HRF), and a binarized representation of the input signal based on the stimulus onsets, the so-called experimental paradigm (EP). However, the EP may not characterize the whole complexity of the activity-inducing signals that evoke the hemodynamic changes. Furthermore, the HRF is known to vary across brain areas and stimuli. To achieve an adaptable framework that can capture such dynamics of the brain function, we model the multivariate fUS time-series as convolutive mixtures and apply block-term decomposition on a set of lagged fUS autocorrelation matrices, revealing both the region-specific HRFs and the source signals that induce the hemodynamic responses. We test our approach on two mouse-based fUS experiments. In the first experiment, we present a single type of visual stimulus to the mouse, and deconvolve the fUS signal measured within the mouse brain’s lateral geniculate nucleus, superior colliculus and visual cortex. We show that the proposed method is able to recover back the time instants at which the stimulus was displayed, and we validate the estimated region-specific HRFs based on prior studies. In the second experiment, we alter the location of the visual stimulus displayed to the mouse, and aim at differentiating the various stimulus locations over time by identifying them as separate sources.

Coupled Tensor Block Term Decomposition with Superpixel-Based Graph Laplacian Regularization for Hyperspectral Super-Resolution

Hyperspectral image (HSI) super-resolution aims at improving the spatial resolution of HSI by fusing a high spatial resolution multispectral image (MSI). To preserve local submanifold structures in HSI super-resolution, a novel superpixel graph-based super-resolution method is proposed. Firstly, the MSI is segmented into superpixel blocks to form two-directional feature tensors, then two graphs are created using spectral–spatial distance between the unfolded feature tensors. Secondly, two graph Laplacian terms involving underlying BTD factors of high-resolution HSI are developed, which ensures the inheritance of the spatial geometric structures. Finally, by incorporating graph Laplacian priors with the coupled BTD degradation model, a HSI super-resolution model is established. Experimental results demonstrate that the proposed method achieves better fused results compared with other advanced super-resolution methods, especially on the improvement of the spatial structure.

A fast sketching-based algorithm for rank-[formula omitted] block term decomposition

The rank-(L,L,1) block term decomposition (BTD) of the tensor has recently received increasing attention in diverse high-order data processing, e.g., hyperspectral image restoration and blind source separation. However, the standard alternating least squares algorithm for the rank-(L,L,1) BTD (BTD-ALS) is computationally expensive, which hinders its application on real large-scale data. In this paper, we propose a fast sketching-based algorithm for the rank-(L,L,1) BTD (FastBTD) to address the computational burden of BTD-ALS. Since the dominant cost of BTD-ALS at each iteration is the sequence of large-scale least squares subproblems, we first project these original subproblems into the low-dimensional subspace by the tensor sketching operator. Then, we can fastly obtain the approximate solutions of the original subproblems by solving these sketched small-scale subproblems at each iteration of FastBTD. The computational complexity of FastBTD (O(2JILR+JIR)) is significantly lower than that of BTD-ALS (O(2I3LR+I3R)) for solving the least squares subproblems at each iteration, when J≪I2. Moreover, we provide the theoretical error bound for FastBTD. Extensive experiments on both synthetic and real data demonstrate that FastBTD achieves substantial speedup while maintaining accuracy compared with BTD-ALS.

Block term decomposition with distinct time granularities for temporal knowledge graph completion

To solve the incompleteness problem in temporal knowledge graphs (TKGs) and to discover the new knowledge, TKG completion remains an essential task always solved by graph embedding technology. Existing TKG completion methods encode time at only single granularity, which is insufficient in exploiting the rich information of distinct time granularities. Furthermore, most of them lack a comprehensive consideration of the characteristic of both time points and time periods, resulting in the inability to handle the two types of facts with different time forms, namely the discrete facts and continuous facts, simultaneously. In this paper, we propose a novel TKG embedding model which introduces the block term tensor decomposition and utilizes the core tensor and factor matrices to capture information presented by facts under distinct time granularities. By focusing on moments included in the time period and treating the discrete fact as a special case of the continuous fact, the model manages the processing of different types of facts in a unified manner. Besides, we explicitly design the static properties of entities and relations as well as their interactions to conform to reality. Experiments on 3 real datasets of different types verify the effectiveness of our proposed method compared with most state-of-the-art methods.

Blind Source Separation in Persistent Atrial Fibrillation Electrocardiograms Using Block-Term Tensor Decomposition With Löwner Constraints.

The estimation of the atrial activity (AA) signal from electrocardiogram (ECG) recordings is an important step in the noninvasive analysis of atrial fibrillation (AF), the most common sustained arrhythmia encountered in clinical practice. This problem admits a blind source separation (BSS) formulation that has been recently posed as a tensor factorization, using the Hankel-based block term decomposition (BTD), which is particularly well-suited to the estimation of exponential models like AA during AF. However, persistent forms of AF are characterized by short R-R intervals and very disorganized (or weak) AA, making it difficult to model AA directly and perform its successful extraction through Hankel-BTD. To overcome this drawback, the present work proposes a tensor approach to estimate QRS complexes and subtract them from the ECG, resulting in a signal that, ideally, only contains the AA component. Such an approach tackles the problem of blind separation of rational functions, which models QRS complexes explicitly. The data tensor admitting a BTD is built from Löwner matrices generated from each lead of the observed ECG. To this end, this paper formulates a variant of the recently proposed constrained alternating group lasso (CAGL) algorithm that imposes Löwner structure on the decomposition blocks. This is done by performing an orthogonal projection, which we explicitly derive, at each iteration of CAGL. Results from experiments with synthetic data show the consistency of the proposed Löwner-constrained AGL (LCAGL) in extracting the desired sources. Experimental results obtained on a population of 20 patients suffering from persistent AF show that the proposed variant outperforms other tensor-based methods in terms of atrial signal estimation quality from ECG records as short as a single heartbeat.

Low-Rank Tucker-2 Model for Multi-Subject fMRI Data Decomposition With Spatial Sparsity Constraint.

Tucker decomposition can provide an intuitive summary to understand brain function by decomposing multi-subject fMRI data into a core tensor and multiple factor matrices, and was mostly used to extract functional connectivity patterns across time/subjects using orthogonality constraints. However, these algorithms are unsuitable for extracting common spatial and temporal patterns across subjects due to distinct characteristics such as high-level noise. Motivated by a successful application of Tucker decomposition to image denoising and the intrinsic sparsity of spatial activations in fMRI, we propose a low-rank Tucker-2 model with spatial sparsity constraint to analyze multi-subject fMRI data. More precisely, we propose to impose a sparsity constraint on spatial maps by using an ℓp norm (0 < p ≤ 1), in addition to adding low-rank constraints on factor matrices via the Frobenius norm. We solve the constrained Tucker-2 model using alternating direction method of multipliers, and propose to update both sparsity and low-rank constrained spatial maps using half quadratic splitting. Moreover, we extract new spatial and temporal features in addition to subject-specific intensities from the core tensor, and use these features to classify multiple subjects. The results from both simulated and experimental fMRI data verify the improvement of the proposed method, compared with four related algorithms including robust Kronecker component analysis, Tucker decomposition with orthogonality constraints, canonical polyadic decomposition, and block term decomposition in extracting common spatial and temporal components across subjects. The spatial and temporal features extracted from the core tensor show promise for characterizing subjects within the same group of patients or healthy controls as well.

Block-Term Tensor Decomposition Model Selection and Computation: The Bayesian Way

The so-called block-term decomposition (BTD) tensor model, especially in its rank-$(L_r,L_r,1)$ version, has been recently receiving increasing attention due to its enhanced ability of representing systems and signals that are composed of \emph{blocks} of rank higher than one, a scenario encountered in numerous and diverse applications. Uniqueness conditions and fitting methods have thus been thoroughly studied. Nevertheless, the challenging problem of estimating the BTD model structure, namely the number of block terms, $R$, and their individual ranks, $L_r$, has only recently started to attract significant attention, mainly through regularization-based approaches which entail the need to tune the regularization parameter(s). In this work, we build on ideas of sparse Bayesian learning (SBL) and put forward a fully automated Bayesian approach. Through a suitably crafted multi-level \emph{hierarchical} probabilistic model, which gives rise to heavy-tailed prior distributions for the BTD factors, structured sparsity is \emph{jointly} imposed. Ranks are then estimated from the numbers of blocks ($R$) and columns ($L_r$) of non-negligible energy. Approximate posterior inference is implemented, within the variational inference framework. The resulting iterative algorithm completely avoids hyperparameter tuning, which is a significant defect of regularization-based methods. Alternative probabilistic models are also explored and the connections with their regularization-based counterparts are brought to light with the aid of the associated maximum a-posteriori (MAP) estimators. We report simulation results with both synthetic and real-word data, which demonstrate the merits of the proposed method in terms of both rank estimation and model fitting as compared to state-of-the-art relevant methods.

Grassmannian Optimization for Online Tensor Completion and Tracking With the t-SVD

We propose a new fast streaming algorithm for the tensor completion problem of imputing missing entries of a low-tubal-rank tensor using the tensor singular value decomposition (t-SVD) algebraic framework. We show the t-SVD is a specialization of the well-studied block-term decomposition for third-order tensors, and we present an algorithm under this model that can track changing free submodules from incomplete streaming 2-D data. The proposed algorithm uses principles from incremental gradient descent on the Grassmann manifold of subspaces to solve the tensor completion problem with linear complexity and constant memory in the number of time samples. We provide a local expected linear convergence result for our algorithm. Our empirical results are competitive in accuracy but much faster in compute time than state-of-the-art tensor completion algorithms on real applications to recover temporal chemo-sensing and MRI data under limited sampling.

A Blind Block Term Decomposition of High Order Tensors

Tensor decompositions have found many applications in signal processing, data mining, machine learning, etc. In particular, the block term decomposition (BTD), which is a generalization of CP decomposition and Tucker decomposition/HOSVD, has been successfully used for the compression and acceleration of neural networks. However, computing BTD is NP-hard, and optimization based methods usually suffer from slow convergence or even fail to converge, which limits the applications of BTD. This paper considers a “blind” block term decomposition (BBTD) of high order tensors, in which the block diagonal structure of the core tensor is unknown. Our contributions include: 1) We establish the necessary and sufficient conditions for the existence of BTD, characterize the condition when a BTD solves the BBTD problem, and show that the BBTD is unique under a “low rank” assumption. 2) We propose an algebraic method to compute the BBTD. This method transforms the problem of determining the block diagonal structure of the core tensor into a clustering problem of complex numbers, in polynomial time. And once the clustering problem is solved, the BBTD can be obtained via computing several matrix decompositions. Numerical results show that our method is able to compute the BBTD, even in the presence of noise to some extent, whereas optimization based methods (e.g., MINF and NLS in TENSORLAB) may fail to converge.

Structuring Data with Block Term Decomposition: Decomposition of Joint Tensors and Variational Block Term Decomposition as a Parametrized Mixture Distribution Model

Structuring Data with Block Term Decomposition: Decomposition of Joint Tensors and Variational Block Term Decomposition as a Parametrized Mixture Distribution Model