Block Term Decomposition Research Articles

Bilinear factorizations subject to monomial equality constraints via tensor decompositions

The Canonical Polyadic Decomposition (CPD), which decomposes a tensor into a sum of rank one terms, plays an important role in signal processing and machine learning. In this paper we extend the CPD framework to the more general case of bilinear factorizations subject to monomial equality constraints. This includes extensions of multilinear algebraic uniqueness conditions originally developed for the CPD. We obtain a deterministic uniqueness condition that admits a constructive interpretation. Computationally, we reduce the bilinear factorization problem into a CPD problem, which can be solved via a matrix EigenValue Decomposition (EVD). Under the given conditions, the discussed EVD-based algorithms are guaranteed to return the exact bilinear factorization. Finally, we make a connection between bilinear factorizations subject to monomial equality constraints and the coupled block term decomposition, which allows us to translate monomial structures into low-rank structures.

Krylov-Levenberg-Marquardt Algorithm for Structured Tucker Tensor Decompositions

Structured Tucker tensor decomposition models complete or incomplete multiway data sets (tensors), where the core tensor and the factor matrices can obey different constraints. The model includes block-term decomposition or canonical polyadic decomposition as special cases. We propose a very flexible optimization method for the structured Tucker decomposition problem, based on the second-order Levenberg-Marquardt optimization, using an approximation of the Hessian matrix by the Krylov subspace method. An algorithm with limited sensitivity of the decomposition is included. The proposed algorithm is shown to perform well in comparison to existing tensor decomposition methods.

Block-Term Tensor Decomposition: Model Selection and Computation

The so-called block-term decomposition (BTD) tensor model 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. Its uniqueness and approximation have thus been thoroughly studied. Nevertheless, the challenging problem of estimating the BTD model structure, namely the number of block terms and their individual ranks, has only recently started to attract significant attention. In this paper, a novel method of BTD model selection and computation is proposed, based on the idea of imposing column sparsity \emph{jointly} on the factors and in a \emph{hierarchical} manner and estimating the ranks as the numbers of factor columns of non-negligible magnitude. Following a block successive upper bound minimization (BSUM) approach for the proposed optimization problem is shown to result in an alternating hierarchical iteratively reweighted least squares (HIRLS) algorithm, which is fast converging and enjoys high computational efficiency, as it relies in its iterations on small-sized sub-problems with closed-form solutions. Simulation results for both synthetic examples and a hyper-spectral image de-noising application are reported, which demonstrate the superiority of the proposed scheme over the state-of-the-art in terms of success rate in rank estimation as well as computation time and rate of convergence.

Systems of Polynomial Equations, Higher-Order Tensor Decompositions, and Multidimensional Harmonic Retrieval: A Unifying Framework. Part II: The Block Term Decomposition

In Part I we proposed a multilinear algebra framework to solve 0-dimensional systems of polynomial equations with simple roots. We extend this framework to incorporate multiple roots: a block term decomposition (BTD) of the null space of the Macaulay matrix reveals the dual (sub)space of a disjoint root in each term. The BTD is the joint triangularization of multiplication tables and a three-way generalization of the Jordan canonical form in the matrix case, intimately related to the border rank of a tensor. We hint at and illustrate flexible numerical optimization-based algorithms.

Nonlocal Block-Term Decomposition for Hyperspectral Image Mixed Noise Removal

Since the facility restrictions and weather conditions, hyperspectral image (HSI) is generally seriously polluted by a variety of noises. Recently, the method based on block-term decomposition with rank- $(L, L, 1)$ (BTD) has attracted wide attention in HSI mixed noise removal. BTD factorizes third-order HSI data into the sum of a series of component tensors, where each of the component tensors is represented by the outer product of a rank- $L$ matrix $\mathbf {A}_r\mathbf {B}_r^T$ and a column vector $\mathbf {c}_r$ . BTD has clear physical interpretation because its latent factors $\mathbf {A}_r\mathbf {B}_r^T$ and $\mathbf {c}_r$ can be interpreted abundance map and spectral signature, respectively. The essential uniqueness of BTD is under the low-rank assumption of $\mathbf {A}_r\mathbf {B}_r^T$ . However, the low-rank assumption is not always held in real scenarios. The BTD-based method usually sets $L$ to full rank to achieve satisfactory results. In this article, we suggest a novel model based on nonlocal block-term decomposition (NLBTD) for HSI mixed noise removal. More specifically, for each grouped similar image block, BTD is introduced to capture nonlocal self-similarity and global spectral low-rankness, the unidirectional total variation is introduced to preserve local spectral smoothness. By faithfully exploring nonlocal self-similarity, global spectral low-rankness, and local spectral smoothness, the proposed method is expected to produce promising results with guarantee the essential uniqueness of BTD. To tackle the resulting model, we design an efficient algorithm based on the proximal alternating minimization with the theoretical guarantees. Extensive numerical experiments in HSI mixed noise removal demonstrate that the proposed NLBTD method achieves satisfactory performance compared with state-of-the-art methods.

Hyperspectral Super-Resolution via Interpretable Block-Term Tensor Modeling

This work revisits coupled tensor decomposition (CTD)-based hyperspectral super-resolution (HSR). HSR aims at fusing a pair of hyperspectral and multispectral images to recover a super-resolution image (SRI). The vast majority of the HSR approaches take a low-rank matrix recovery perspective. The challenge is that theoretical guarantees for recovering the SRI using low-rank matrix models are either elusive or derived under stringent conditions. A couple of recent CTD-based methods ensure recoverability for the SRI under relatively mild conditions, leveraging on algebraic properties of the canonical polyadic decomposition (CPD) and the Tucker decomposition models, respectively. However, the latent factors of both the CPD and Tucker models have no physical interpretations in the context of spectral image analysis, which makes incorporating prior information challenging---but using priors is often essential for enhancing performance in noisy environments. This work employs an idea that models spectral images as tensors following the block-term decomposition model with multilinear rank-$(L_r, L_r, 1)$ terms (i.e., the LL1 model) and formulates the HSR problem as a coupled LL1 tensor decomposition problem. Similar to the existing CTD approaches, recoverability of the SRI is shown under mild conditions. More importantly, the latent factors of the LL1 model can be interpreted as the key constituents of spectral images, i.e., the endmembers' spectral signatures and abundance maps. This connection allows us to easily incorporate prior information for performance enhancement. A flexible algorithmic framework that can work with a series of structural information is proposed to take advantage of the model interpretability. The effectiveness is showcased using simulated and real data.

Hyperspectral Restoration via $L_0$ Gradient Regularized Low-Rank Tensor Factorization

Due to the mechanism of the data acquisition process, hyperspectral imagery (HSI) are usually contaminated by various noises, e.g., Gaussian noise, impulse noise, strips, and dead lines. In this article, a spectral–spatial $L_{0}$ gradient regularized low-rank tensor factorization (LRTF $L_{0}$ ) method is proposed for hyperspectral denoising, in which the restored HSI is approximated by low-rank block term decomposition (BTD). BTD factorizes a tensor into the sum of a series of component tensors, each of which is represented by the outer product of a matrix and a vector. From subspace learning point of view, the vector and matrix can be considered as a spectral atom and its corresponding coding coefficients. In the proposed method, the correlations in both spectral and spatial domains are taken into account via the small size of atom set and low-rankness of coding matrices. In addition, HSIs also have the local structure of piecewise smoothness in both spectral and spatial domains. Motivated by the supreme virtues of $L_{0}$ gradient regularization in image structure exploitation, we develop a spectral–spatial $L_{0}$ gradient regularization and embed it into BTD to explore the spectral–spatial texture information. The proposed method can simultaneously remove various types of noises, and the experimental results on both synthetic data and real-world data show its superiority when compared with several state-of-the-art approaches.

Tensor Decomposition for Multilayer Networks Clustering

Clustering on multilayer networks has been shown to be a promising approach to enhance the accuracy. Various multilayer networks clustering algorithms assume all networks derive from a latent clustering structure, and jointly learn the compatible and complementary information from different networks to excavate one shared underlying structure. However, such an assumption is in conflict with many emerging real-life applications due to the existence of noisy/irrelevant networks. To address this issue, we propose Centroid-based Multilayer Network Clustering (CMNC), a novel approach which can divide irrelevant relationships into different network groups and uncover the cluster structure in each group simultaneously. The multilayer networks is represented within a unified tensor framework for simultaneously capturing multiple types of relationships between a set of entities. By imposing the rank-(Lr,Lr,1) block term decomposition with nonnegativity, we are able to have well interpretations on the multiple clustering results based on graph cut theory. Numerically, we transform this tensor decomposition problem to an unconstrained optimization, thus can solve it efficiently under the nonlinear least squares (NLS) framework. Extensive experimental results on synthetic and real-world datasets show the effectiveness and robustness of our method against noise and irrelevant data.

Tensor Based Blind Source Separation in Longitudinal Magnetic Resonance Imaging Analysis.

Study of white matter (WM) fiber-bundles is a crucial challenge in the investigation of neurological diseases like multiple sclerosis (MS). In this activity, the amount of data to process is huge, and an automated approach to carry out this task is in order.In this paper we show how tensor-based blind source separation (BSS) techniques can be successfully applied to model complex anatomical brain structures. More in detail, we show how through vector hankelization it is possible to formalize data extracted from WM fiber-bundles using a tensor model. Two main tensor factorization techniques, namely (Lr, Lr, 1) block term decomposition (BTD) and canonical polyadic decomposition (CPD), were applied to the generated tensor. The information extracted from the factorization was then used to differentiate between sets of fibers, within the bundle, affected by the pathology and normal appearing fibers.Performances of the proposed tensor-based model was evaluated on simulated data representing pathological effects of MS. Results show the capability of our tensor-based model to detect small pathological phenomena appearing along WM fibers.

BTDP

Bilinear models are very powerful in multimodal fusion tasks like Visual Question Answering. The predominant bilinear methods can all be seen as a kind of tensor-based decomposition operation that contains a key kernel called “core tensor.” Current approaches usually focus on reducing the computation complexity by applying low-rank constraint on the core tensor. In this article, we propose a novel bilinear architecture called Block Term Decomposition Pooling (BTDP), which not only maintains the advantages of previous bilinear methods but also conducts sparse bilinear interactions between modalities. Our method is based on Block Term Decompositions theory of tensor, which will result in a sparse and learnable block-diagonal core tensor for multimodal fusion. We prove that using such a block-diagonal core tensor is equivalent to conducting many “tiny” bilinear operations in different feature spaces. Thus, introducing sparsity into the bilinear operation can significantly increase the performance of feature fusion and improve VQA models. What is more, our BTDP is very flexible in design. We develop several variants of BTDP and discuss the effects of the diagonal blocks of core tensor. Extensive experiments on two challenging VQA-v1 and VQA-v2 datasets show that our BTDP method outperforms current bilinear models, achieving state-of-the-art performance.

Weighted Channel-Wise Decomposed Convolutional Neural Networks

Currently, block term decomposition is widely utilized to factorize regular convolutional kernels with several groups to decrease parameters. However, networks designed based on this method lack adequate information interactions from every group. Therefore, the Weighted Channel-wise Decomposed Convolutions (WCDC) are proposed in this paper, and the relevant networks can be called WCDC-Nets. The WCDC convolutional kernel employ the channel-wise decomposition to reduce the parameters and computational complexity to the bone. Furthermore, a tiny learnable weighted module is also utilized to dig up connections of the outputs from channel-wise convolutions in the WCDC kernel. The WCDC filter can be easily applied in many popular networks and can be trained end to end, resulting in a significant improvement of model’s flexibility. Experimental results on the benchmark datasets showed that WCDC-Nets can achieve better performances with much fewer parameters and flop pointing computations.

Block Term Decomposition of ECG Recordings for Atrial Fibrillation Analysis: Temporal and Inter-Patient Variability

Responsible for 25% of strokes and 1/3 of hospitalizations due to cardiac related disturbances, atrial fibrillation (AF) is the most common sustained cardiac arrhythmia in clinical practice, considered as the last great frontier of cardiac electrophysiology. Its mechanisms are not completely understood, and a precise analysis of the atrial activity (AA) signal in electrocardiogram (ECG) recordings is necessary to better understand this challenging cardiac condition. Recently, the block term decomposition (BTD) has been proposed as a powerful tool to noninvasively extract AA in AF ECG signals. However, this tensor factorization technique was performed only in short ECG recordings and illustrated in single patients. To assess its performance and variability through different subjects, BTD is applied to a population of 10 AF patients in this paper. Also, its time variability is evaluated by means of long segments of AF ECG with varying observation window size. Experimental results show the consistency of BTD as an AA extraction tool, outperforming two well-known matrix-based methods in most of the observed cases for long and short AF ECG recordings.

Blind fMRI source unmixing via higher-order tensor decompositions

BackgroundThe growing interest in neuroimaging technologies generates a massive amount of biomedical data of high dimensionality. Tensor-based analysis of brain imaging data has been recognized as an effective analysis that exploits its inherent multi-way nature. In particular, the advantages of tensorial over matrix-based methods have previously been demonstrated in the context of functional magnetic resonance imaging (fMRI) source localization. However, such methods can also become ineffective in realistic challenging scenarios, involving, e.g., strong noise and/or significant overlap among the activated regions. Moreover, they commonly rely on the assumption of an underlying multilinear model generating the data. New methodThis paper aims at investigating the possible gains from exploiting the 4-dimensional nature of the brain images, through a higher-order tensorization of the fMRI signal, and the use of less restrictive generative models. In this context, the higher-order block term decomposition (BTD) and the PARAFAC2 tensor models are considered for the first time in fMRI blind source separation. A novel PARAFAC2-like extension of BTD (BTD2) is also proposed, aiming at combining the effectiveness of BTD in handling strong instances of noise and the potential of PARAFAC2 to cope with datasets that do not follow the strict multilinear assumption. Comparison with existing methodsThe methods were tested using both synthetic and real data and compared with state of the art methods. ConclusionsThe simulation results demonstrate the effectiveness of BTD and BTD2 for challenging scenarios (presence of noise, spatial overlap among activation regions and inter-subject variability in the haemodynamic response function (HRF)).

On the minimal ranks of matrix pencils and the existence of a best approximate block-term tensor decomposition

Under the action of the general linear group with tensor structure, the ranks of matrices A and B forming an m×n pencil A+λB can change, but in a restricted manner. Specifically, with every pencil one can associate a pair of minimal ranks, which is unique up to a permutation. This notion can be defined for matrix pencils and, more generally, also for matrix polynomials of arbitrary degree. In this paper, we provide a formal definition of the minimal ranks, discuss its properties and the natural hierarchy it induces in a pencil space. Then, we show how the minimal ranks of a pencil can be determined from its Kronecker canonical form. For illustration, we classify the orbits according to their minimal ranks (under the action of the general linear group) in the case of real pencils with m,n≤4. Subsequently, we show that real regular 2k×2k pencils having only complex-valued eigenvalues, which form an open positive-volume set, do not admit a best approximation (in the norm topology) on the set of real pencils whose minimal ranks are bounded by 2k−1. Our results can be interpreted from a tensor viewpoint, where the minimal ranks of a degree-(d−1) matrix polynomial characterize the minimal ranks of matrices constituting a block-term decomposition of an m×n×d tensor into a sum of matrix-vector tensor products.

Nonconvulsive Epileptic Seizure Detection in Scalp EEG Using Multiway Data Analysis.

Nonconvulsive status epilepticus is a condition where the patient is exposed to abnormally prolonged epileptic seizures without evident physical symptoms. Since thesecontinuous seizures may cause permanent brain damage, it constitutes a medical emergency. This paper proposes a method to detect nonconvulsive seizures for a further nonconvulsive status epilepticus diagnosis. To differentiate between the normal and seizure electroencephalogram (EEG), a K-Nearest Neighbor, a Radial Basis Support Vector Machine, and a Linear Discriminant Analysis classifier are used. The classifier features are obtained from the Canonical Polyadic Decomposition (CPD) and Block Term Decomposition of the EEG data represented as third order tensor. To expand the EEG into a tensor, Wavelet or Hilbert-Huang transform are used. The algorithm is tested on a scalp EEG database of 139 seizures of different duration. The experimental results suggest that a Hilbert-Huang tensor representation and the CPD analysis provide the most suitable framework for nonconvulsive seizure detection. The Radial Basis Support Vector Machine classifier shows the best performance with sensitivity, specificity, and accuracy values over 98%. A rough comparison with other methods proposed in the literature shows the superior performance of the proposed method for nonconvulsive epileptic seizure detection.

The User Behavior Analysis Based on Text Messages Using Parafac and Block Term Decomposition

Tensor decompositions represent a start for big data analysis and a start in reduction of dimensionality, object detection, clustering and so on. This paper presents a method to study the behavior of users in the online environment and beyond. A beginning for analyzing this type of data is uniting the Parafac Tensor Decomposition and the Block Term Decomposition.

DeepSearch

Content-based image retrieval (CBIR) is one of the most important applications of computer vision. In recent years, there have been many important advances in the development of CBIR systems, especially Convolutional Neural Networks (CNNs) and other deep-learning techniques. On the other hand, current CNN-based CBIR systems suffer from high computational complexity of CNNs. This problem becomes more severe as mobile applications become more and more popular. The current practice is to deploy the entire CBIR systems on the server side while the client side only serves as an image provider. This architecture can increase the computational burden on the server side, which needs to process thousands of requests per second. Moreover, sending images have the potential of personal information leakage. As the need of mobile search expands, concerns about privacy are growing. In this article, we propose a fast image search framework, named DeepSearch, which makes complex image search based on CNNs feasible on mobile phones. To implement the huge computation of CNN models, we present a tensor Block Term Decomposition (BTD) approach as well as a nonlinear response reconstruction method to accelerate the CNNs involving in object detection and feature extraction. The extensive experiments on the ImageNet dataset and Alibaba Large-scale Image Search Challenge dataset show that the proposed accelerating approach BTD can significantly speed up the CNN models and further makes CNN-based image search practical on common smart phones.

Blind Multichannel Deconvolution and Convolutive Extensions of Canonical Polyadic and Block Term Decompositions

Tensor decompositions such as the canonical polyadic decomposition (CPD) or the block term decomposition (BTD) are basic tools for blind signal separation. Most of the literature concerns instantaneous mixtures/memoryless channels. In this paper, we focus on convolutive extensions. More precisely, we present a connection between convolutive CPD/BTD models and coupled but instantaneous CPD/BTD. We derive a new identifiability condition dedicated to convolutive low-rank factorization problems. We explain that under this condition, the convolutive extension of CPD/BTD can be computed by means of an algebraic method, guaranteeing perfect source separation in the noiseless case. In the inexact case, the algorithm can be used as a cheap initialization for an optimization-based method. We explain that, in contrast to the memoryless case, convolutive signal separation is in certain cases possible despite only two-way diversities (e.g., space $\times$ time).

Non-orthogonal tensor diagonalization

Tensor diagonalization means transforming a given tensor to an exactly or nearly diagonal form through multiplying the tensor by non-orthogonal invertible matrices along selected dimensions of the tensor. It has a link to an approximate joint diagonalization (AJD) of a set of matrices. In this paper, we derive (1) a new algorithm for a symmetric AJD, which is called two-sided symmetric diagonalization of an order-three tensor, (2) a similar algorithm for a non-symmetric AJD, also called a two-sided diagonalization of an order-three tensor, and (3) an algorithm for three-sided diagonalization of order-three or order-four tensors. The latter two algorithms may serve for canonical polyadic (CP) tensor decomposition, and in certain scenarios they can outperform traditional CP decomposition methods. Finally, we propose (4) similar algorithms for tensor block diagonalization, which is related to tensor block-term decomposition. The proposed algorithm can either outperform the existing block-term decomposition algorithms, or produce good initial points for their application.

Matrix-Vector Nonnegative Tensor Factorization for Blind Unmixing of Hyperspectral Imagery

Many spectral unmixing approaches ranging from geometry, algebra to statistics have been proposed, in which nonnegative matrix factorization (NMF)-based ones form an important family. The original NMF-based unmixing algorithm loses the spectral and spatial information between mixed pixels when stacking the spectral responses of the pixels into an observed matrix. Therefore, various constrained NMF methods are developed to impose spectral structure, spatial structure, and spectral-spatial joint structure into NMF to enforce the estimated endmembers and abundances preserve these structures. Compared with matrix format, the third-order tensor is more natural to represent a hyperspectral data cube as a whole, by which the intrinsic structure of hyperspectral imagery can be losslessly retained. Extended from NMF-based methods, a matrix-vector nonnegative tensor factorization (NTF) model is proposed in this paper for spectral unmixing. Different from widely used tensor factorization models, such as canonical polyadic decomposition CPD) and Tucker decomposition, the proposed method is derived from block term decomposition, which is a combination of CPD and Tucker decomposition. This leads to a more flexible frame to model various application-dependent problems. The matrix-vector NTF decomposes a third-order tensor into the sum of several component tensors, with each component tensor being the outer product of a vector (endmember) and a matrix (corresponding abundances). From a formal perspective, this tensor decomposition is consistent with linear spectral mixture model. From an informative perspective, the structures within spatial domain, within spectral domain, and cross spectral-spatial domain are retreated interdependently. Experiments demonstrate that the proposed method has outperformed several state-of-the-art NMF-based unmixing methods.