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.

We consider tensor data completion of an incomplete observation of multidimensional harmonic (MH) signals. Unlike existing tensor-based techniques for MH retrieval (MHR), which mostly adopt the canonical polyadic decomposition (CPD) to model the simple "one-to-one" correspondence among harmonics across difference modes, we herein use the more flexible block term decomposition (BTD) model that can be used to describe the complex mutual correspondences among several groups of harmonics across different modes. An optimization principle that aims to fit the BTD model in the least squares sense, subject to rank minimization of hankelized MH components, is set up for the tensor completion task, and an algorithm based on alternating direction method of multipliers is proposed, of which the effectiveness and applicability are validated through both numerical simulations and an application in sub-6GHz channel state information (CSI) completion.

Tucker decomposition has shown advantages in simultaneously extracting group shared and individual features for studying brain function from multi-subject fMRI data. However, Tucker decomposition of complex-valued fMRI data is challenging, since the data are highly noisy, and imposing sparsity constraints on spatial maps, previously used for denoising magnitude-only fMRI data, may remove signal voxels with low amplitudes. Here we propose a new complex-valued low-rank Tucker decomposition (clrTKD) method to extract principal group and individual components from multi-subject fMRI data, and to denoise spatial components based on the small phase change property at a post-processing stage. We derive the update rules using the alternating direction method of multipliers. Simulated and experimental complex-valued fMRI data are used to evaluate the proposed clrTKD method. Results show that the proposed method can extract more contiguous and vital brain activations such as the anterior cingulate cortex region, compared to Tucker decomposition for magnitude-only fMRI data.

Adaptive canonical polyadic decomposition (CPD) is a technique for tracking time-varying tensor models in real time or at low cost, and has attracted much interest across a variety of signal processing applications. However, most existing CPD trackers assume slow variations of the model with fixed dynamic pattern, and some key parameters, such as the forgetting factor that weights the importance of historical observed data, remain unchanged during the tracking process. Therefore, without monitoring the state of the dynamic CPD model, these techniques may be less performant or even fail in cases where the model suddenly varies dramatically at some point, or the dynamic pattern of the model changes over time. To address this limitation, we propose a hybrid "STATe monitoring assisted TRAcKing" (STATTRAK) method, integrating a state monitor into the tracking process, based on which the CPD tracker adaptively selects the tracking (or decomposition) strategy as well as the forgetting factor. Note that the state monitor evaluates the variation of the dynamic CPD model between adjacent time points, in a recursive way, and thus is also of low-cost. This work focuses on third-order adaptive CPD, where data is added along one mode and factor matrices in the other two modes have fixed size with time-varying entries. Experiments show that STATTRAK consistently achieves higher accuracy than existing CPD trackers and significantly reduces CPU time compared with batch methods. Specifically, it is at least 15.5% more accurate than the compared CPD trackers, in terms of RMSE reduction, and is at least 14 times faster than batch decomposition, in terms of CPU time.

This paper considers target localization with a multistatic MIMO radar system of multiple transmit arrays and multiple receive arrays. We formulate the matched filtered output data into tensors that admit the double coupled canonical polyadic decomposition (DC-CPD) model, which efficiently characterizes the coupling between receive arrays and that between transmit arrays, and multilinear structure in each tensor. We propose a novel algebraic DC-CPD algorithm based on coupled rank-1 detection mapping, and provides analysis into the uniqueness conditions. A post-processing approach is also introduced to calculate and fuse the DOD and DOA information from the DC-CPD results to finally obtain the target locations. Simulations are provided to illustrate the merits of proposed method over CPD and C-CPD methods, with regards to accuracy and identifiability.

Coupled canonical polyadic decomposition (C-CPD) of multiple tensors is a fundamental tool for multi-set data fusion. Existing C-CPD works are mainly limited to batch processing techniques for stationary models, yet in practice the C-CPD model may be dynamic and thus adaptive C-CPD tracking techniques are in urgent need. In this paper, we consider the problem of adaptive tracking of a time-varying third-order C-CPD model, and propose a recursive least squares based adaptive tracking algorithm. Theoretical and experimental results are provided to show the merits of the proposed C-CPD tracking algorithm over batch C-CPD algorithm and CPD tracking algorithm, in terms of improved accuracy, reduced complexity, and more relaxed working conditions.

Independence and sparsity are proved to be two basic features for spatial activations of functional magnetic resonance imaging (fMRI) data, and have shown efficiency in analysis of magnitude-only fMRI data. Since complex-valued fMRI data contains additional brain activity information beyond magnitude-only fMRI data, we propose to incorporate sparsity constraint into complex independent vector analysis (IVA) to take advantages of the two features in analyzing multi-subject complex-valued fMRI data. Specifically, we propose to improve a complex-valued IVA algorithm named AFIVA (adaptive fixed-point IVA) to add a phase sparsity constraint on spatial maps. Based on the cost function of AFIVA, we further implement the phase sparsity constraint using smoothed Lo norm, and utilize noncircularity of spatial maps as well in the second update of phase sparsity to extract meaningful activations. The results from experimental complex-valued fMRI datasets show that the proposed method yields higher accuracy than AFIV A in terms of true positive rates, confirming the advantage of sparsity in de-noising the independent spatial maps.

The decomposition of multi-subject fMRI data using rank-(L,L,1,1) block term decomposition (BTD) can preserve higher-way data structure and is more robust to noise effects by decomposing shared spatial maps (SMs) into a product of two rank-L loading matrices. However, since the number of whole-brain voxels is very large and rank L is larger than 1, the rank-(L,L,1,1) BTD requires high computation and memory. Therefore, we propose an accelerated rank-(L,L,1,1) BTD algorithm based upon the method of alternating least squares (ALS). We speed up updates of loading matrices by reducing fMRI data into subspaces, and add an orthonormality constraint on shared SMs to improve the performance. Moreover, we evaluate the rank-L effect on the proposed method for actual task-related fMRI data. The proposed method shows better performance when L=35. Meanwhile, experimental comparison results verify that the proposed method largely reduced (17.36 times) computation time compared to ALS while also providing satisfying separation performance.

Convolutional neural networks (CNNs) have shown promising results in classifying individuals with mental disorders such as schizophrenia using resting-state fMRI data. However, complex-valued fMRI data is rarely used since additional phase data introduces high-level noise though it is potentially useful information for the context of classification. As such, we propose to use spatial source phase (SSP) maps derived from complex-valued fMRI data as the CNN input. The SSP maps are not only less noisy, but also more sensitive to spatial activation changes caused by mental disorders than magnitude maps. We build a 3D-CNN framework with two convolutional layers (named SSPNet) to fully explore the 3D structure and voxel-level relationships from the SSP maps. Two interpretability modules, consisting of saliency map generation and gradient-weighted class activation mapping (Grad-CAM), are incorporated into the well-trained SSPNet to provide additional information helpful for understanding the output. Experimental results from classifying schizophrenia patients (SZs) and healthy controls (HCs) show that the proposed SSPNet significantly improved accuracy and AUC compared to CNN using magnitude maps extracted from either magnitude-only (by 23.4 and 23.6% for DMN) or complex-valued fMRI data (by 10.6 and 5.8% for DMN). SSPNet captured more prominent HC-SZ differences in saliency maps, and Grad-CAM localized all contributing brain regions with opposite strengths for HCs and SZs within SSP maps. These results indicate the potential of SSPNet as a sensitive tool that may be useful for the development of brain-based biomarkers of mental disorders.

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$ \ell _{p} $norm (${0}< {p}\le {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.

Tucker decomposition (TKD) has been utilized to identify functional connectivity patterns using processed fMRI data, but seldom focuses on originally acquired fMRI data. This study proposes to decompose multi-subject fMRI data in a natural three-way of voxel × time × subject via TKD. Different from existing tensor decomposition algorithms such as canonical polyadic decomposition (CPD) for extracting shared spatial maps (SMs), we propose to extract both shared and individual SMs by exploring spatial-temporal-subject relationship contained in the core tensor. We test the proposed method using multi-subject resting-state fMRI data with comparison to CPD for evaluating shared SMs and independent vector analysis (IVA) for assessing individual SMs under different model orders. The results show that the proposed method yields better and more robust shared SMs than CPD and more consistent individual SMs than IVA, indicating the potential of TKD in providing group and individual brain networks in a high-dimensional coupling way.

Spatial source phase (SSP), derived from complex-valued functional magnetic resonance imaging (fMRI) data by data-driven methods, has unique capacity of identifying blood oxygenation-level dependent (BOLD)-related voxels from noisy voxels regardless of their amplitudes. However, the use of SSP constraint in sparse representation algorithms have rarely been studied. This study proposes a sparse representation method using SSP hard thresholding to achieve the sparsity of spatial components, enabling the use of initially complex-valued fMRI data and retaining the brain information embedded in noisy voxels and weak BOLD-related voxels with small phase values. Rank-1 matrix estimation is applied to sequentially update dictionary atoms and corresponding spatial components, followed by hard thresholding on spatial components based on SSP. The proposed method is evaluated using both simulated and experimental complex-valued data. The results show that the proposed method yields better performance than a complex-valued dictionary learning algorithm when using initially acquired complex-valued task-related fMRI data.

Convolutional neural network (CNN) can achieve better classification by using independent component analysis (ICA) components derived from complex-valued fMRI data than from magnitude-only fMRI data due to incorporating additional phase information. However, thus far magnitude slices of only a single brain network (i.e. spatial component) has been used in the classification. This study aims to take advantages of multiple ICA components in providing rich information and to provide a conclusion for efficient multiple-component fusion. More precisely, we present three fusion approaches: 1) averaging multiple ICA components as inputs of a single-component CNN, 2) concatenating features of multiple single-component CNN, and 3) averaging predictive probabilities of multiple single-component CNN. We evaluate the proposed methods using resting-state fMRI data collected from 42 schizophrenia patients and 40 healthy controls. Experimental results show that all three fusion approaches can improve classification accuracy compared to the single-component CNN, and the first approach performs the best. No matter which fusion method is used, we reach the same conclusion that four-component fusion is sufficient to obtain satisfying performance, and two-component fusion yields higher improvement and better performance than the single-component classification, especially when using components having good accuracy for single-component CNN classification.

Canonical polyadic decomposition (CPD) of multi-subject complex-valued fMRI data can be used to provide spatially and temporally shared components among groups with both magnitude and phase information. However, the CPD model is not well formulated due to the large subject variability in the spatial and temporal modalities, as well as the high noise level in complexvalued fMRI data. Considering that the shift-invariant CPD can model temporal variability across subjects, we propose to further impose a phase sparsity constraint on the shared spatial maps to denoise the complex-valued components and to model the inter-subject spatial variability as well. More precisely, subject-specific time delays are first estimated for the complex-valued shared time courses in the framework of real-valued shift-invariant CPD. Source phase sparsity is then imposed on the complex-valued shared spatial maps. A smoothed ℓ0norm is specifically used to reduce voxels with large phase values after phase de-ambiguity based on the small phase characteristic of BOLD-related voxels. The results from both the simulated and experimental fMRI data demonstrate improvements of the proposed method over three complex-valued algorithms, namely, tensor-based spatial ICA, shift-invariant CPD and CPD without spatiotemporal constraints. When comparing with a real-valued algorithm combining shiftinvariant CPD and ICA, the proposed method detects 178.7% more contiguous task-related activations.

Magnitude-only resting-state fMRI data have been largely investigated via independent component analysis (ICA) for exacting spatial maps (SMs) and time courses. However, the native complex-valued fMRI data have rarely been studied. Motivated by the significant improvements achieved by ICA of complex-valued task fMRI data than magnitude-only task fMRI data, we present an efficient method for de-noising SM estimates which makes full use of complex-valued resting-state fMRI data. Our two main contributions include: (1) The first application of a post-ICA phase de-noising method, originally proposed for task fMRI data, to resting-state data, which recognizes voxels within a specific phase range as desired voxels. (2) A new phase range detection strategy for a specific SM component based on correlation with its reference. We continuously change the phase range within a larger range, and compute a set of correlation coefficients between each de-noised SM and its reference. The phase range with the maximal correlation determines the final selection. The detected results by the proposed approach confirm the correctness of the post-ICA phase de-noising method in the analysis of resting-state complex-valued fMRI data.

Coupled decompositions of multiple tensors are fundamental tools for multi-set data fusion. In this paper, we introduce a coupled version of the rank-(Lm, Ln, •) block term decomposition (BTD), applicable to joint independent subspace analysis. We propose two algorithms for its computation based on a coupled block simultaneous generalized Schur decomposition scheme. Numerical results are given to show the performance of the proposed algorithms.

Independent vector analysis (IVA) has exhibited great potential for the group analysis of magnitude-only fMRI data, but has rarely been applied to native complex-valued fMRI data. We propose an adaptive fixed-point IVA algorithm by taking into account the extremely noisy nature, large variability of the source component vector (SCV) distribution, and non-circularity of the complex-valued fMRI data. The multivariate generalized Gaussian distribution (MGGD) is exploited to match the SCV distribution based on nonlinearity, the shape parameter of MGGD is estimated using maximum likelihood estimation, and the nonlinearity is updated in the dominant SCV subspace to achieve denoising goal. In addition, the pseudo-covariance matrix is incorporated into the algorithm to represent the non-circularity. Experimental results from simulated and actual fMRI data demonstrate significant improvements of our algorithm over a complex-valued IVA-G algorithm and several circular and noncircular fixed-point IVA variants.

This study presents a method to analyze blood-oxygen-level-dependent (BOLD) functional magnetic resonance imaging (fMRI) signals associated with listening to continuous music. Semi-blind independent component analysis (ICA) was applied to decompose the fMRI data to source level activation maps and their respective temporal courses. The unmixing matrix in the source separation process of ICA was constrained by a variety of acoustic features derived from the piece of music used as the stimulus in the experiment. This allowed more stable estimation and extraction of more activation maps of interest compared to conventional ICA methods.

Complex blind source separation (BSS) received growing interests in many practical applications in the past decades, and non-orthogonal joint diagonalization (JD) of a set of complex matrices plays an instrumental role in solving these problems. In this paper, we propose a new complex non-orthogonal JD algorithm. This algorithm successively finds the optimal Givens and hyperbolic rotation matrices that constitute the elementary rotation matrix in each iteration in an alternating manner. It does not require the target matrices to be Hermitian, and thus could be well adapted to BSS problems that involve fourth-order cumulant slices or time-lagged covariance matrices. Simulations are provided to compare the proposed algorithm with other JD algorithms.

Overcomplete independent component analysis (ICA) is a challenge of ICA to estimate more sources from less mixtures. The statistical properties of the sources such as sparsity are often assumed to solve the problem. Other available information about the sources such as waveform, however, is scarcely used. Motivated by the fact that semi-blind ICA in complete case can improve the potential of ICA by incorporating source information, this paper proposes a semi-blind algorithm for overcomplete ICA by explicitly utilizing waveform information about some sources. An approximate expectation-maximization (EM) algorithm is explored to provide normal cost function of the semi-blind algorithm while the prior information is utilized to form an extended one. Computer simulations results demonstrate that the proposed algorithm has much improved performance in SNR, convergence speed, and elimination of order ambiguity compared to the original EM algorithm.