Canonical Polyadic Decomposition Research Articles

RTensor: An R Package for Multidimensional Array (Tensor) Unfolding, Multiplication, and Decomposition

rTensor is an R package designed to provide a common set of operations and decompositions for multidimensional arrays (tensors). We provide an S4 class that wraps around the base 'array' class and overloads familiar operations to users of 'array', and we provide additional functionality for tensor operations that are becoming more relevant in recent literature. We also provide a general unfolding operation, for which the k-mode unfolding and the matrix vectorization are special cases of. Finally, package rTensor implements common tensor decompositions such as canonical polyadic decomposition, Tucker decomposition, multilinear principal component analysis, t-singular value decomposition, as well as related matrix-based algorithms such as generalized low rank approximation of matrices and popular value decomposition.

The Condition Number of Join Decompositions

The join set of a finite collection of smooth embedded submanifolds of a mutual vector space is defined as their Minkowski sum. Join decompositions generalize some ubiquitous decompositions in multilinear algebra, namely tensor rank, Waring, partially symmetric rank and block term decompositions. This paper examines the numerical sensitivity of join decompositions to perturbations; specifically, we consider the condition number for general join decompositions. It is characterized as a distance to a set of ill-posed points in a supplementary product of Grassmannians. We prove that this condition number can be computed efficiently as the smallest singular value of an auxiliary matrix. For some special join sets, we characterized the behavior of sequences in the join set converging to the latter's boundary points. Finally, we specialize our discussion to the tensor rank and Waring decompositions and provide several numerical experiments confirming the key results.

Damage localization in composite plates using canonical polyadic decomposition of Lamb wave difference signals tensor

Monitoring in real-time and autonomously the health state of aeronautic structures is referred to as Structural Health Monitoring (SHM) and is a process decomposed in four steps: damage detection, localization, classification, and quantification. Structures under study are here composite structures representative of aeronautic applications and the focus is put on the localization step of the SHM process. The fact that SHM data are naturally three-way tensors is here investigated for this purpose. It is demonstrated that under classical assumptions regarding wave propagation, the canonical polyadic decomposition of rank 2 of the tensor built from the phase of the difference signals between a healthy and damaged states provides direct access to the distances between the piezoelectric elements and the damage. This property is used here to propose an original and robust tensor-based damage localization algorithm. This algorithm is successfully validated on experimental data coming from composite plates with mounted piezoelectric elements and compared with a classical localization algorithm based on triangulation.

A Riemannian Trust Region Method for the Canonical Tensor Rank Approximation Problem

The canonical tensor rank approximation problem (TAP) consists of approximating a real-valued tensor by one of low canonical rank, which is a challenging non-linear, non-convex, constrained optimization problem, where the constraint set forms a non-smooth semi-algebraic set. We introduce a Riemannian Gauss-Newton method with trust region for solving small-scale, dense TAPs. The novelty of our approach is threefold. First, we parametrize the constraint set as the Cartesian product of Segre manifolds, hereby formulating the TAP as a Riemannian optimization problem, and we argue why this parametrization is among the theoretically best possible. Second, an original ST-HOSVD-based retraction operator is proposed. Third, we introduce a hot restart mechanism that efficiently detects when the optimization process is tending to an ill-conditioned tensor rank decomposition and which often yields a quick escape path from such spurious decompositions. Numerical experiments show improvements of up to three orders of magnitude in terms of the expected time to compute a successful solution over existing state-of-the-art methods.

Rapid acquisition of data dense solid-state CPMG NMR spectral sets using multi-dimensional statistical analysis.

The development of multi-dimensional statistical methods has been demonstrated on variable contact time (VCT) 29Si{1H} cross-polarization magic angle spinning (CP/MAS) data sets collected using Carr-Purcell-Meiboom-Gill (CPMG) type acquisition. These methods utilize the transformation of the collected 2D VCT data set into a 3D data set and use tensor-rank decomposition to extract the spectral components that vary as a function of transverse relaxation time (T2) and CP contact time. The result is a data dense spectral set that can be used to reconstruct CP/MAS spectra at any contact time with a high signal to noise ratio and with an excellent agreement to 29Si{1H} CP/MAS spectra collected using conventional acquisition. These CPMG data can be collected in a fraction of time that would be required to collect a conventional VCT data set. We demonstrate the method on samples of functionalized mesoporous silica materials and show that the method can provide valuable surface specific information about their functional chemistry.

Recommending Mobile Microblog Users via a Tensor Factorization Based on User Cluster Approach

User influence is a very important factor for microblog user recommendation in mobile social network. However, most existing user influence analysis works ignore user’s temporal features and fail to filter the marketing users with low influence, which limits the performance of recommendation methods. In this paper, a Tensor Factorization based User Cluster (TFUC) model is proposed. We firstly identify latent influential users by neural network clustering. Then, we construct a features tensor according to latent influential user’s opinion, activity, and network centrality information. Furthermore, user influences are predicted by the latent factors resulting from the temporal restrained CP decomposition. Finally, we recommend microblog users considering both user influence and content similarity. Our experimental results show that the proposed model significantly improves recommendation performance. Meanwhile, the mean average precision of TFUC outperforms the baselines with 3.4% at least.

A new class of block coordinate algorithms for the joint eigenvalue decomposition of complex matrices

Several signal processing problems can be written as the joint eigenvalue decomposition (JEVD) of a set of noisy matrices. JEVD notably occurs in source separation problems and for the canonical polyadic decomposition of tensors. Most of the existing JEVD algorithms are based on a block coordinate procedure and require significant modifications to deal with complex-valued matrices. These modifications decrease algorithms performances either in terms of estimation accuracy of the eigenvectors or in terms of computational cost. Therefore, we propose a class of algorithms working equally with real- or complex-valued matrices. These algorithms are still based on a block coordinate procedure and multiplicative updates. The originality of the proposed approach lies in the structure of the updating matrix and in the way the optimization problem is solved in CN×N. That structure is parametrized and allows to define up to five different JEVD algorithms. Thanks to numerical simulations, we show that, with respect to the more accurate algorithms of the literature, this approach improves the estimation of the eigenvectors and has a computational cost significantly lower. Finally, as an application example, one of the proposed algorithm is successfully applied to the blind source separation of Direct-Sequence Code Division Multiple Access signals.

Tensor-structured coupled cluster theory.

We derive and implement a new way of solving coupled cluster equations with lower computational scaling. Our method is based on the decomposition of both amplitudes and two electron integrals, using a combination of tensor hypercontraction and canonical polyadic decomposition. While the original theory scales as O(N6) with respect to the number of basis functions, we demonstrate numerically that we achieve sub-millihartree difference from the original theory with O(N4) scaling. This is accomplished by solving directly for the factors that decompose the cluster operator. The proposed scheme is quite general and can be easily extended to other many-body methods.

Non-orthogonal Simultaneous Diagonalization of K-Order Complex Tensors for Source Separation

Source separation in the statistical framework is usually managed using tensor decompositions or matrix joint diagonalization. In this letter, we propose one of the first coordinate algorithms for non-orthogonal simultaneous diagonalization of any order complex tensors. It relies on the optimization of an inverse criterion and on a particular decomposition allowing to derive each parameter in an independent way. In the framework of digital telecommunication source separation, computer simulations show the interest of using sets of high-order tensors. They also illustrate the overall interesting performances of the proposed algorithm in comparison to a Jacobi-like algorithm of matrix joint diagonalization and to a canonical polyadic decomposition algorithm.

TensorBeat

Breathing signal monitoring can provide important clues for health problems. Compared to existing techniques that require wearable devices and special equipment, a more desirable approach is to provide contact-free and long-term breathing rate monitoring by exploiting wireless signals. In this article, we propose TensorBeat, a system to employ channel state information (CSI) phase difference data to intelligently estimate breathing rates for multiple persons with commodity WiFi devices. The main idea is to leverage the tensor decomposition technique to handle the CSI phase difference data. The proposed TensorBeat scheme first obtains CSI phase difference data between pairs of antennas at the WiFi receiver to create CSI tensors. Then canonical polyadic (CP) decomposition is applied to obtain the desired breathing signals. A stable signal matching algorithm is developed to identify the decomposed signal pairs, and a peak detection method is applied to estimate the breathing rates for multiple persons. Our experimental study shows that TensorBeat can achieve high accuracy under different environments for multiperson breathing rate monitoring.

Condition numbers for the tensor rank decomposition

The tensor rank decomposition problem consists of recovering the parameters of the model from an identifiable low-rank tensor. These parameters are analyzed and interpreted in many applications. As tensors are often perturbed by measurement errors in practice, one must investigate to what extent the unique parameters change in order to preserve the validity of the analysis. The magnitude of this change can be bounded asymptotically by the product of the condition number and the norm of the perturbation to the tensor. This paper introduces a condition number that admits a closed expression as the inverse of a particular singular value of Terracini's matrix, which represents the tangent space to the set of tensors of fixed rank. A practical algorithm for computing this condition number is presented. The latter's elementary properties such as scaling and orthogonal invariance are established. Rank-1 tensors are always well-conditioned. The class of weak 3-orthogonal tensors, which includes orthogonally decomposable tensors, contains both well-conditioned and ill-conditioned problems. Numerical experiments confirm that the condition number yields a good estimate of the magnitude of the change of the parameters when the tensor is perturbed.

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).

Automatic detection of T wave alternans using tensor decompositions in multilead ECG signals

Objective: T wave alternans (TWA) is a promising non-invasive risk stratification tool for sudden cardiac death which can be detected from surface ECG. This paper proposes a novel method to automatically detect TWA based on tensor decomposition methods. Approach: Two different tensor decomposition approaches are examined and compared, namely canonical polyadic decomposition and the more generalized variation PARAFAC2 which allows the T waves to shift in time. Results and Significance: Results on different artificial and clinical signals show that the presented methods are a robust and reliable way for TWA detection, and show the potential benefit of tensors in ECG signal processing.

Tensor Decomposition for Signal Processing and Machine Learning

Tensors or {\em multi-way arrays} are functions of three or more indices $(i,j,k,\cdots)$ -- similar to matrices (two-way arrays), which are functions of two indices $(r,c)$ for (row,column). Tensors have a rich history, stretching over almost a century, and touching upon numerous disciplines; but they have only recently become ubiquitous in signal and data analytics at the confluence of signal processing, statistics, data mining and machine learning. This overview article aims to provide a good starting point for researchers and practitioners interested in learning about and working with tensors. As such, it focuses on fundamentals and motivation (using various application examples), aiming to strike an appropriate balance of breadth {\em and depth} that will enable someone having taken first graduate courses in matrix algebra and probability to get started doing research and/or developing tensor algorithms and software. Some background in applied optimization is useful but not strictly required. The material covered includes tensor rank and rank decomposition; basic tensor factorization models and their relationships and properties (including fairly good coverage of identifiability); broad coverage of algorithms ranging from alternating optimization to stochastic gradient; statistical performance analysis; and applications ranging from source separation to collaborative filtering, mixture and topic modeling, classification, and multilinear subspace learning.

Fault Detection Algorithms based on Decomposed Tensor Representations for Qualitative Models

The paper proposes two fault detection algorithms for qualitative models based on stochastic automata. We will show that storing the transition probabilities of the stochastic automaton in tensor format enables a great potential in avoiding the major limitation of the approach - the exponential growth of the number of transitions of the automaton with an increasing number of system signals. The underlying structure of the behaviour tensor of the automaton will be exploited by CP and TT decompositions which allow a reduction in the amount of data to be stored by an order of magnitude. We will provide a proof of both algorithms and show their functionality by means of a real system example.

A new canonical polyadic decomposition algorithm with improved stability and its applications to biomedical signal processing

In biomedical engineering applications, high-order data, also known as tensors, are more and more popular and canonical polyadic decomposition (CPD) is one of the most powerful tool to analyze such high-dimensional high-order data. However, existing CPD algorithms suffer from a serious disadvantage: they are prone to stick into local minima and hence may result in unreasonable components that are hard to interpret. To overcome this problem, we proposed a new CPD algorithm that not only provides significantly improved stability but is also very suitable for parallel computing. The performance of the proposed algorithm was justified by using synthetic data and its applications in biomedical image denoising.

Target Detection for Polarized Hyperspectral Images Based on Tensor Decomposition

In the past decades, with the development of polarized hyperspectral imaging system, target detection and recognition for polarized hyperspectral images (PHSIs) have attracted more and more attention. Most conventional target detection algorithms of PHSI are based on the Stokes vector, which mainly take advantage of the spectral information, and ignore the continuous variability of polarized dimension, being similar to spectrum. In fact, the PHSI include multidimensional features of polarization, spectrum, space, and radiation, and these provide more discriminable information about target and background than traditional spectrum or intensity ones. Hence, tensor, which can keep the complete information of PHSI, is introduced to represent such high-dimensional data. In this letter, the tensor canonical polyadic (CP) decomposition is adopted to extract the spectral and polarized features. There are two ways to realize the target detection through CP decomposition. One is to construct a fourth-order tensor matched filtering (FTMF), and FTMF is applied to the original data directly without extracting the Stokes vector. The other is to reconstruct the PHSI into a new 3-D data, then matched filter algorithm of hyperspectral image is applied to this data to detect the targets, for short CPMF. The experimental results show that the proposed methods achieve the joint utilization of spectrum and polarization and are more suitable and effective for the target detection of the PHSI.

On identifiability of 3-tensors of multilinear rank $(1,\ L_{r},\ L_{r})$

In this paper, we study a specific big data model via multilinear rank tensor decompositions. The model approximates to a given tensor by the sum of multilinear rank $(1, L_{r}, L_{r})$ terms. And we characterize the identifiability property of this model from a geometric point of view. Our main results consists of exact identifiability and generic identifiability. The arguments of generic identifiability relies on the exact identifiability, which is in particular closely related to the well-known trisecant lemma in the context of algebraic geometry (see Proposition 2.6 in [ 1 ]). This connection discussed in this paper demonstrates a clear geometric picture of this model.

A Type-2 Block-Component-Decomposition Based 2D AOA Estimation Algorithm for an Electromagnetic Vector Sensor Array.

This paper investigates a two-dimensional angle of arrival (2D AOA) estimation algorithm for the electromagnetic vector sensor (EMVS) array based on Type-2 block component decomposition (BCD) tensor modeling. Such a tensor decomposition method can take full advantage of the multidimensional structural information of electromagnetic signals to accomplish blind estimation for array parameters with higher resolution. However, existing tensor decomposition methods encounter many restrictions in applications of the EMVS array, such as the strict requirement for uniqueness conditions of decomposition, the inability to handle partially-polarized signals, etc. To solve these problems, this paper investigates tensor modeling for partially-polarized signals of an L-shaped EMVS array. The 2D AOA estimation algorithm based on rank- BCD is developed, and the uniqueness condition of decomposition is analyzed. By means of the estimated steering matrix, the proposed algorithm can automatically achieve angle pair-matching. Numerical experiments demonstrate that the present algorithm has the advantages of both accuracy and robustness of parameter estimation. Even under the conditions of lower SNR, small angular separation and limited snapshots, the proposed algorithm still possesses better performance than subspace methods and the canonical polyadic decomposition (CPD) method.

High-order possibilistic c-means algorithms based on tensor decompositions for big data in IoT

Internet of Things (IoT) connects the physical world and the cyber world to offer intelligent services by data mining for big data. Each big data sample typically involves a large number of attributes, posing a remarkable challenge on the high-order possibilistic c-means algorithm (HOPCM). Specially, HOPCM requires high-performance servers with a large-scale memory and a powerful computing unit, to cluster big samples, limiting its applicability in IoT systems with low-end devices such as portable computing units and embedded devises which have only limited memory space and computing power. In this paper, we propose two high-order possibilistic c-means algorithms based on the canonical polyadic decomposition (CP-HOPCM) and the tensor-train network (TT-HOPCM) for clustering big data. In detail, we use the canonical polyadic decomposition and the tensor-train network to compress the attributes of each big data sample. To evaluate the performance of our algorithms, we conduct the experiments on two representative big data datasets, i.e., NUS-WIDE-14 and SNAE2, by comparison with the conventional high-order possibilistic c-means algorithm in terms of attributes reduction, execution time, memory usage and clustering accuracy. Results imply that CP-HOPCM and TT-HOPCM are potential for big data clustering in IoT systems with low-end devices since they can achieve a high compression rate for heterogeneous samples to save the memory space significantly without a significant clustering accuracy drop.