Canonical Polyadic Decomposition Research Articles

Small Target Detection in Infrared Videos Based on Spatio-Temporal Tensor Model

Existing methods of the small target detection from infrared videos are not effective with the complex background. It is mainly caused by: 1) the interference of strong edges and the similarity with other nontarget objects and 2) the lack of the context information of both the background and the target in a spatio-temporal domain. By considering these two points, we propose to slide a window in a single frame and form a spatio-temporal cube with the current frame patch and other frame patches in the spatio-temporal domain. Then, we establish a spatio-temporal tensor model based on these patches. According to the sparse prior of the target and the local correlation of the background, the separation of the target and the background can be cast as a low rank and sparse tensor decomposition problem. The target is obtained from the sparse tensor by the tensor decomposition. The experiments show that our method gains better detection performance in infrared videos with the complex background by making full use of the spatio-temporal context information.

Tensor-Based Channel Estimation for Millimeter Wave MIMO-OFDM With Dual-Wideband Effects

We consider the channel estimation problem in millimeter wave (mmWave) multiple-input multiple-output orthogonal frequency division multiplexing (MIMO-OFDM) systems with hybrid analog-digital architectures. Leveraging the spatial- and frequency-wideband (dual-wideband) effects in massive MIMO scenarios, we derive a spatial-frequency channel model with dual-wideband effects that incorporates the multipath parameters, i.e., time delay, complex gain, angle of departure/arrival. We adopt a successive beam training scheme and formulate the training OFDM signal as a third-order low-rank tensor fitting a canonical polyadic (CP) model with factor matrices containing the channel parameters. Exploiting the Vandermonde nature of factor matrices, we propose a structured CP decomposition-based channel estimation strategy aided by the spatial smoothing method, where two dedicated algorithms with particular tensor modeling and parameter recovery operations are developed. The proposed scheme leverages standard linear algebra, and, hence, avoids the random initialization problem and iterative procedure. An analysis of the uniqueness condition of CP decomposition is also pursued. Simulation results indicate that the proposed strategy achieves enhanced estimation performance, which outperforms the traditional approaches in terms of accuracy, robustness and complexity.

Canonical polyadic decomposition (CPD) of big tensors with low multilinear rank

Tensor decomposition methods have been widely applied to big data analysis as they bring multiple modes and aspects of data to a unified framework, which allows us to discover complex internal structures and correlations of data. Unfortunately most existing approaches are not designed to meet the challenges posed by big data dilemma. This paper attempts to improve the scalability of tensor decompositions and makes two contributions: A flexible and fast algorithm for the CP decomposition (FFCP) of tensors based on their Tucker compression; A distributed randomized Tucker decomposition approach for arbitrarily big tensors but with relatively low multilinear rank. These two algorithms can deal with huge tensors, even if they are dense. Extensive simulations provide empirical evidence of the validity and efficiency of the proposed algorithms.

Coupled CP Decomposition of Simultaneous MEG-EEG Signals for Differentiating Oscillators During Photic Driving.

Magnetoencephalography (MEG) and electroencephalography (EEG) are contemporary methods to investigate the function and organization of the brain. Simultaneously acquired MEG-EEG data are inherently multi-dimensional and exhibit coupling. This study uses a coupled tensor decomposition to extract the signal sources from MEG-EEG during intermittent photic stimulation (IPS). We employ the Coupled Semi-Algebraic framework for approximate CP decomposition via SImultaneous matrix diagonalization (C-SECSI). After comparing its performance with alternative methods using simulated benchmark data, we apply it to MEG-EEG recordings of 12 participants during IPS with fractions of the individual alpha frequency between 0.4 and 1.3. In the benchmark tests, C-SECSI is more accurate than SECSI and alternative methods, especially in ill-conditioned scenarios, e.g., involving collinear factors or noise sources with different variances. The component field-maps allow us to separate physiologically meaningful oscillations of visually evoked brain activity from background signals. The frequency signatures of the components identify either an entrainment to the respective stimulation frequency or its first harmonic, or an oscillation in the individual alpha band or theta band. In the group analysis of both, MEG and EEG data, we observe a reciprocal relationship between alpha and theta band oscillations. The coupled tensor decomposition using C-SECSI is a robust, powerful method for the extraction of physiologically meaningful sources from multidimensional biomedical data. Unsupervised signal source extraction is an essential solution for rendering advanced multi-modal signal acquisition technology accessible to clinical diagnostics, pre-surgical planning, and brain computer interface applications.

Multi-population mortality forecasting using tensor decomposition

ABSTRACT In this paper, we formulate the multi-population mortality forecasting problem based on 3-way (age, year, and country/gender) decompositions. By applying the canonical polyadic decomposition (CPD) and the different forms of the Tucker decomposition to multi-population mortality data (10 European countries and 2 genders), we find that the out-of-sample forecasting performance is significantly improved both for individual populations and the aggregate population compared with using the single-population mortality model based on rank-1 singular value decomposition (SVD), or the Lee–Carter model. The results also shed lights on the similarity and difference of mortality among different countries. Additionally, we compare the variance-explained method and the out-of-sample validation method for rank (hyper-parameter) selection. Results show that the out-of-sample validation method is preferred for forecasting purposes.

A unitary joint diagonalization algorithm for nonsymmetric higher‐order tensors based on Givens‐like rotations

SummaryBased on Givens‐like rotations, we present a unitary joint diagonalization algorithm for a set of nonsymmetric higher‐order tensors. Each unitary rotation matrix only depends on one unknown parameter which can be analytically obtained in an independent way following a reasonable assumption and a complex derivative technique. It can serve for the canonical polyadic decomposition of a higher‐order tensor with orthogonal factors. Furthermore, based on cross‐high‐order cumulants of observed signals, we show that the proposed algorithm can be applied to solve the joint blind source separation problem. The simulation results reveal that the proposed algorithm has a competitive performance compared with those of several existing related methods.

CP decomposition and weighted clique problem

The determination of the optimal subdecomposition based on CP decomposition for high-order tensors is not straightforward because the rank-one tensors are not orthogonal. We show that this is equivalent to the weighted clique problem. Thus, it is NP-hard.

Krylov subspace projection method for Sylvester tensor equation with low rank right-hand side

Motivated by the effectiveness of Krylov projection methods and the CP decomposition of tensors, which is a low rank decomposition, we propose Arnoldi-based methods (block and global) to solve Sylvester tensor equation with low rank right-hand sides. We apply a standard Krylov subspace method to each coefficient matrix, in order to reduce the main problem to a projected Sylvester tensor equation, which can be solved by a global iterative scheme. We show how to extract approximate solutions via matrix Krylov subspaces basis. Several theoretical results such as expressions of residual and its norm are presented. To show the performance of the proposed approaches, some numerical experiments are given.

Crop characterization using an improved scattering power decomposition technique for compact polarimetric SAR data

Crop characterization using Compact-Pol Synthetic Aperture Radar (CP-SAR) data is of prime interest with the rapid advancements of SAR systems towards operational applications. It is noteworthy that as a good compromise between the dual and quad-polarized SAR systems, the CP-SAR offer advantages in terms of the larger swath and lower data rate. The m − χ CP decomposition considers two out of the three Stokes child parameters: degree of polarization (m), ellipticity (χ), and orientation angle (ψ) to describe the polarized part of the quasi-monochromatic partially polarized wave. An improvement in the scattering powers was proposed in the S − Ω decomposition, which takes into accounts both the transmitted and received wave ellipticities (χt, χr) and the orientation angles (ψt, ψr). In this decomposition, S denotes the Stokes vector and Ω is the polarized power fraction. However, it may be noted that the S − Ω decomposition intrinsically ignores dominance in the target scattering mechanism while calculating the powers. In this work, improvement is proposed for the S − Ω decomposition by utilizing the degree of dominance in the scattering mechanism. The improved S − Ω (named as iS − Ω) decomposition powers are first compared with the existing m − χ and S − Ω powers for elementary (viz., trihedral and dihedral corner reflectors) and distributed targets using simulated CP-SAR data from quad-pol RADARSAT-2 data. An increase of ∼2% for odd and even-bounce powers obtained from the iS − Ω decomposition is observed for the trihedral and dihedral corner reflectors respectively. The analysis of the scattering powers for distributed targets shows that an increase of 15% and 12% in the even and odd-bounce powers is observed from iS − Ω for urban and bare soil areas respectively as compared to the m − χ and S − Ω decompositions. Besides, temporal variations in the scattering powers obtained from the iS − Ω decomposition are also analyzed for rice, cotton, and sugarcane crops at different growth stages.

Tensor train representation of MIMO channels using the JIRAFE method

MIMO technology has been subject of increasing interest in both academia and industry for future wireless standards. However, its performance benefits strongly depend on the accuracy of the channel at the base station. In a recent work, a fourth-order channel tensor model was proposed for MIMO systems. In this paper, we extend this model by exploiting additional spatial diversity at the receiver, which induces a fifth order tensor model for the channel. For such high orders, there is a crucial need to break the initial high-dimensional optimization problem into a collection of smaller coupled optimization sub-problems. This paper exploits new results on the equivalence between the canonical polyadic decomposition (CPD) and the tensor train (TT) decomposition for the multi-path scenario. Specifically, we propose a Joint dImensionality Reduction And Factor rEtrieval (JIRAFE) method to find the transmit and receive spatial signatures as well as the complex path gains (which also capture the polarization effects). Monte Carlo simulations show that our proposed TT-based representation of the channel is more robust to noise and computationally more attractive than available competing tensor-based methods, for physical parameters estimation.

Using the proximal gradient and the accelerated proximal gradient as a canonical polyadic tensor decomposition algorithms in difficult situations

Canonical Polyadic (CP) tensor decomposition is useful in many real-world applications due to its uniqueness, and the ease of interpretation of its factor matrices. This work addresses the problem of calculating the CP decomposition of tensors in difficult cases where the factor matrices in one or all modes are almost collinear – i.e. bottleneck or swamp problems arise. This is done by introducing a constraint on the coherences of the factor matrices that ensures the existence of a best low-rank approximation, which makes it possible to estimate these highly correlated factors. Two new algorithms optimizing the CP decomposition based on proximal methods are proposed. Simulation results are provided and demonstrate the good behaviour of these algorithms, as well as a better compromise between accuracy and convergence speed than other algorithms in the literature.

Transforming high-dimensional potential energy surfaces into a canonical polyadic decomposition using Monte Carlo methods.

A Monte Carlo method is proposed for transforming high-dimensional potential energy surfaces evaluated on discrete grid points into a sum-of-products form, more precisely into a Canonical Polyadic Decomposition form. To this end, a modified existing ansatz based on the alternating least squares method is used, in which numerically exact integrals are replaced with Monte Carlo integrals. This largely reduces the numerical cost by avoiding the evaluation of the potential on all grid points and allows the treatment of surfaces with many degrees of freedom. Calculations on the 15D potential of the protonated water dimer (Zundel cation) in a sum-of-products form are presented and compared to the results obtained in a previous work [M. Schröder and H.-D. Meyer, J. Chem. Phys. 147, 064105 (2017)], where a sum-of-products form of the potential was obtained in the Tucker format.

Multi-Set Low-Rank Factorizations With Shared and Unshared Components

Low-rank matrix/tensor factorizations play a significant role in science and engineering. An important example is the canonical polyadic decomposition (CPD). There is also a growing interest in multi-set extensions of low-rank matrix/tensor factorizations in which the associated factor matrices are partially shared. In this paper we propose a more unified framework for multi-set matrix/tensor factorizations. In particular, we propose a multi-set extension of bilinear factorizations subject to monomial equality constraints to the case of shared and unshared factors. The presented framework encompasses (generalized) canonical correlation analysis (CCA) and (coupled) CPD models as special cases. CPD, CCA and hybrid models between them feature interesting uniqueness properties. We derive uniqueness conditions for CCA and multi-set low-rank factorization with partially shared entities. Computationally, we reduce multi-set low-rank factorizations with shared and unshared components into a special CPD problem, which can be solved via a matrix eigenvalue decomposition. Finally, numerical experiments demonstrate the importance of taking the coupling between multi-set low-rank factorizations into account in the actual computation.

Block-Randomized Stochastic Proximal Gradient for Low-Rank Tensor Factorization

This work considers the problem of computing the canonical polyadic decomposition (CPD) of large tensors. Prior works mostly leverage data sparsity to handle this problem, which is not suitable for handling dense tensors that often arise in applications such as medical imaging, computer vision, and remote sensing. Stochastic optimization is known for its low memory cost and per-iteration complexity when handling dense data. However, exisiting stochastic CPD algorithms are not flexible enough to incorporate a variety of constraints/regularizations that are of interest in signal and data analytics. Convergence properties of many such algorithms are also unclear. In this work, we propose a stochastic optimization framework for large-scale CPD with constraints/regularizations. The framework works under a doubly randomized fashion, and can be regarded as a judicious combination of randomized block coordinate descent (BCD) and stochastic proximal gradient (SPG). The algorithm enjoys lightweight updates and small memory footprint. In addition, this framework entails considerable flexibility---many frequently used regularizers and constraints can be readily handled under the proposed scheme. The approach is also supported by convergence analysis. Numerical results on large-scale dense tensors are employed to showcase the effectiveness of the proposed approach.

A Second-Order Method for Fitting the Canonical Polyadic Decomposition With Non-Least-Squares Cost

The canonical polyadic decomposition (CPD) can be used to extract meaningful components from a tensor. Most existing optimization methods for fitting the CPD use as cost function the least-squares distance between the tensor and its CPD. While the minimum of this cost function coincides with the maximum likelihood estimator for data with additive i.i.d. Gaussian distributed noise, for other noise distributions, better-suited cost functions exist. For such cost functions, first-order, gradient-based optimization methods have been proposed. However, (approximate) second-order methods, which additionally use information from the Hessian of the cost function to achieve faster convergence, are still largely unexplored. In this paper, we generalize the Gauss–Newton nonlinear least-squares algorithm to twice differentiable entry-wise cost functions. The low-rank structure of the problem is exploited to keep the computational cost low. As a special case, $\beta$ -divergence cost functions are examined. We show that quadratic convergence can be obtained close to the solution with a reasonable extra cost in memory and computation time, making the proposed method particularly useful when high accuracy of the decomposition is desired.

On Recoverability of Randomly Compressed Tensors With Low CP Rank

Our interest lies in the recoverability properties of compressed tensors under the \textit{canonical polyadic decomposition} (CPD) model. The considered problem is well-motivated in many applications, e.g., hyperspectral image and video compression. Prior work studied this problem under somewhat special assumptions---e.g., the latent factors of the tensor are sparse or drawn from absolutely continuous distributions. We offer an alternative result: We show that if the tensor is compressed by a subgaussian linear mapping, then the tensor is recoverable if the number of measurements is on the same order of magnitude as that of the model parameters---without strong assumptions on the latent factors. Our proof is based on deriving a \textit{restricted isometry property} (R.I.P.) under the CPD model via set covering techniques, and thus exhibits a flavor of classic compressive sensing. The new recoverability result enriches the understanding to the compressed CP tensor recovery problem; it offers theoretical guarantees for recovering tensors whose elements are not necessarily continuous or sparse.

Multilinear Singular Value Decomposition for Millimeter Wave Channel Parameter Estimation

Fifth generation (5G) cellular standards are set to utilize millimeter wave (mmWave) frequencies, which enable data speeds greater than 10 Gbps and sub-centimeter localization accuracy. These capabilities rely on accurate estimates of the channel parameters, which we define as the angle of arrival, angle of departure, and path distance for each path between the transmitter and receiver. Estimating the channel parameters in a computationally efficient manner poses a challenge because it requires estimation of parameters from a high-dimensional measurement – particularly for multi-carrier systems since each subcarrier must be estimated separately. Additionally, channel parameter estimation must be able to handle hybrid beamforming, which uses a combination of digital and analog beamforming to reduce the number of required analog to digital converters. This paper introduces a channel parameter estimation technique based on the multilinear singular value decomposition (MSVD), a Tucker form tensor analog of the singular value decomposition, for massive multiple input multiple output (MIMO) multi-carrier systems with hybrid beamforming. The MSVD tensor estimation approach is more computationally efficient than methods such as the canonical polyadic decomposition (CPD) and the Tucker form of the MSVD enables paths to be extracted based on signal energy. The algorithms performance is compared to the CPD method and shown to closely match the Cramer-Rao bound (CRB) of channel parameter estimates through simulations. Additionally, limitations of channel parameter estimation and communication waveform effects are studied.

On Uniqueness and Computation of the Decomposition of a Tensor into Multilinear Rank-$(1,L_r,L_r)$ Terms

Canonical Polyadic Decomposition (CPD) represents a third-order tensor as the minimal sum of rank-1 terms. Because of its uniqueness properties the CPD has found many concrete applications in telec...

Hyperspectral Image Classification Based on Mathematical Morphology and Tensor Decomposition

AbstractHyperspectral Image (HSI) classification refers to classifying hyperspectral data into features, where labels are given to pixels sharing the same features, distinguishing the present materials of the scene from one another. Naturally a HSI acquires spectral features of pixels, but spatial features based on neighborhood information are also important, which results in the problem of spectral-spatial classification. There are various ways to account to spatial information, one of which is through Mathematical Morphology, which is explored in this work. A HSI is a third-order data block, and building new spatial diversities may increase this order. In many cases, since pixel-wise classification requires a matrix of pixels and features, HSI data are reshaped as matrices which causes high dimensionality and ignores the multi-modal structure of the features. This work deals with HSI classification by modeling the data as tensors of high order. More precisely, multi-modal hyperspectral data is built and dealt with using tensor Canonical Polyadic (CP) decomposition. Experiments on real HSI show the effectiveness of the CP decomposition as a candidate for classification thanks to its properties of representing the pixel data in a matrix compact form with a low dimensional feature space while maintaining the multi-modality of the data.

Learning Nonnegative Factors From Tensor Data: Probabilistic Modeling and Inference Algorithm

Tensor canonical polyadic decomposition (CPD) with nonnegative factor matrices, which extracts useful latent information from multidimensional data, has found wide-spread applications in various big data analytic tasks. Currently, the implementation of most existing algorithms needs the knowledge of tensor rank. However, this information is practically unknown and difficult to acquire. To address this issue, a probabilistic approach is taken in this paper. Different from previous works, this paper firstly introduces a sparsity-promoting nonnegative Gaussian-gamma prior, based on which a novel probabilistic model for the CPD problem with nonnegative and continuous factors is established. This probabilistic model further enables the derivation of an efficient inference algorithm that accurately learns the nonnegative factors from the tensor data, along with an integrated feature of automatic rank determination. Numerical results using synthetic data and real-world applications are presented to show the remarkable performance of the proposed algorithm.