DOA and Polarization Parameters Estimation by Exploiting Canonical Polyadic Decomposition of Tensors

Probabilistic inference with noisy-threshold models based on a CP tensor decomposition

Learn 2D filter banks are currently being used in high-impact applications such convolutional neural networks, convolutional sparse representations, etc. However such filter banks usually have plentiful filters, each being non-separable, accounting for a large portion of the overall computational cost. In this paper we propose a novel and computationally appealing alternating optimization based algorithm to estimate a linear combination of separable (rank-1) filters to approximate 2D filter banks. Our computational results show that the proposed method can be faster than (state-of-the-art) tensor Canonical Polyadic decomposition (CPD) method to obtain an approximation of comparable accuracy.

The alternating least squares (ALS) method is frequently used for the computation of the canonical polyadic decomposition (CPD) of tensors. It generally gives accurate solutions, but demands much time. A strong alternative to this is the alternating slice-wise diagonalization (ASD) method. It limits its targets only to third-order tensors, and in exchange for this restriction, it fully utilizes a compression technique based on matrix singular value decomposition and consequently achieves high efficiency. In this paper, we propose a new simple algorithm, Reduced ALS, which lies somewhere between ALS and ASD; it employs a similar compression procedure to ASD, but applies it more directly to ALS. Numerical experiments show that Reduced ALS runs as fast as ASD, avoiding instability ASD sometimes exhibits.

The main contribution of our work consists in developing for the first time a method of estimating the direction of arrival (DOA) parameters assuming noncircular and temporally and spatially correlated signals. This new approach, based on a significant enhancement of the two-sided instrumental variable signal subspace fitting (IV-SSF) method, outperforms its classical version in terms of lower bias and error variance. Moreover, it will be shown that our new method is statistically more efficient than the MODE method especially in the case of partly and fully coherent signals where only the extended and the classical two-sided IV-SSF methods are applicable. We also derive an explicit expression for the stochastic Cramér-Rao bound (CRB) of the DOA estimates from temporally and spatially correlated signals generated from noncircular sources. The new CRB is compared to those of circular temporally correlated and noncircular independent and identically distributed signals to show that the CRB obtained assuming both noncircular sources and temporally correlated signals is lower than the CRBs derived considering only one of these two assumptions. This illustrates the potential gain that both noncircularity and temporal correlation provide when considered together. It will also be proven that the difference between the three CRBs increases with the number of snapshots. However, as the signal-to-noise ratio (SNR) increases, the CRBs merge together and decrease linearly. Moreover, at low SNR values it will be shown that temporal correlation is more informative about the unknown DOA parameters than noncircularity. Finally, the CRB derived assuming noncircular and temporally correlated signals depends on the noncircularity rate, the circularity phase separation, and the DOA separation.

In this work, we consider the problem of factoring a third-order tensor into multilinear rank -($L_{r}, L_{r}$, 1) terms. This model, referred to as the rank-($L_{r}, L_{r}$, 1) block-term decomposition (BTD), finds many applications in signal processing, especially blind separation of smooth sources and unmixing spectral-spatial data (e.g., hyper-spectral image). On the other hand, finding latent factors of rank-($L_{r}, L_{r}$, 1) BTD poses a very challenging optimization problem. Some computational tools designed for canonical polyadic decomposition (CPD) (e.g., alternating least squares (ALS) and Levenberg-Marquardt (LM) based algorithms) can be modified to handle rank-($L_{r}, L_{r}$, 1) BTD. Nonetheless, these methods essentially treat rank-($L_{r}, L_{r}$, 1) BTD as a special CPD problem. This raises a number of challenges, since rank -($L_{r}, L_{r}$, 1) BTD can be viewed as a CPD problem with rank-deficient latent factors and high CP rank–and these are known as hard cases for CPD algorithms. In this work, we reformulate the rank-($L_{r}, L_{r}$, 1) BTD problem as a matrix rank-constrained matrix factorization problem. We propose a simple algorithm that combines alternating optimization and projected gradient. This way, the per-iteration complexity is much smaller than those of the ALS and LM-based algorithms. We also show that the algorithm converges to a stationary point of the problem of interest at a sublinear rate–although nonconvex constraints are involved. Numerical experiments are conducted to showcase the effectiveness of our algorithm.

The compressive array method, where a compression matrix is designed to reduce the dimension of the received signal vector, is an effective solution to obtain high estimation performance with low system complexity. While sparse arrays are often used to obtain higher degrees of freedom (DOFs), in this paper, an orthogonal dipole sparse array structure exploiting compressive measurements is proposed to estimate the direction of arrival (DOA) and polarization signal parameters jointly. Based on the proposed structure, we also propose an estimation algorithm using the compressed sensing (CS) method, where the DOAs are accurately estimated by the CS algorithm and the polarization parameters are obtained via the least‐square method exploiting the previously estimated DOAs. Furthermore, the performance of the estimation of DOA and polarization parameters is explicitly discussed through the Cramér‐Rao bound (CRB). The CRB expression for elevation angle and auxiliary polarization angle is derived to reveal the limit of estimation performance mathematically. The difference between the results given in this paper and the CRB results of other polarized reception structures is mainly due to the use of the compression matrix. Simulation results verify that, compared with the uncompressed structure, the proposed structure can achieve higher estimated performance with a given number of channels.

The Canonical Polyadic (CP) decomposition of R-way arrays is a powerful tool in multilinear algebra. Algorithms to compute an approximate CP decomposition from noisy observations are often based on Alternating Least Squares (ALS) which may require a large number of iterations to converge. To avoid this drawback we investigate semi-algebraic approaches that algebraically reformulate the CP decomposition into a set of simultaneous matrix diagonalization (SMD) problems. In particular, we propose a SEmi-algebraic framework for approximate CP decompositions via SImultaneous matrix diagonalization (SMD) and generalized unfoldings (SECSI-GU). SECSI-GU combines the benefits of two existing semi-algebraic approaches based on SMDs: the SECSI framework which selects the model estimate from multiple candidates obtained by solving multiple SMDs and the “Semi-Algebraic Tensor Decomposition” (SALT) algorithm which considers a “generalized” unfolding of the tensor in order to enhance the identifiability for tensors with R > 3 dimensions. The resulting SECSI-GU framework offers a large number of degrees of freedom to flexibly adapt the performance-complexity trade-off. As we show in numerical simulations, it outperforms SECSI and SALT for tensors with R > 3 dimensions.

In this paper, the direction of arrival (DOA) and adaptive antenna array beamforming for the smart antenna applications are realized by using Minimum Variance Distortionless Response (MVDR) algorithm of DOA estimation. The MVDR supported by joint approximate diagonalization of eigenmatrices (JADE) algorithm is used to estimate the direction of arrivals of coherent signals in non-coherent groups. The generalized steering vectors using JADE algorithm are first estimated, and then the spectrum methods are realized to estimate the DOA of each signal. Root mean square error (RMSE) values are computed in different scenarios for the proposed method as well as JADE-MUSIC and the corresponding results are compared. Although both methods resolved the signals correctly, simulation results show that JADE based MVDR has better performances in the cases of low number of antenna elements and snapshots. The DOAs obtained are then processed using LMS adaptive beamforming algorithm to steer the main lobe of the radiation pattern to desired angles (signals) and the nulls to undesired angles (signals). The simulation results also reveal that adaptive beamforming is successfully done by annihilating the effects of undesired signals significantly.

The r-term representation \(\rm{v}={\sum}^{r}_{v=1}\,\,{\bigotimes}^{d}_{j=1}\,\,{\vartheta}_{v}^{(j)}\), i.e., a representation by sums of r elementary tensors, is already used in the algebraic definition (3.11) of tensors. In different fields, the r-term representation has different names: ‘canonical decomposition’ in psychometrics (cf. [30]), ‘parallel factors model’ (cf. [96]) in chemometrics.1 The word ‘representation’ is often replaced by ‘format’. The short form ‘CP’ is proposed by Comon [38] meaning ‘canonical polyadic decomposition’. Here, the notation ‘r-term representation’ is used with ‘r’ considered as a variable from \(\mathbb{N}_0\), which may be replaced by other variable names or numbers.

Direction-of-arrival (DoA) estimation is essential in many industrial internet of things (IIoT) applications where IoT users may require a high-resolution estimation algorithm for features limited to positioning and device localization. The conventional DoA estimation is challenging as the number of antenna arrays increases leading to an increase in design complexity and cost in RF circuitry. As a result, the conventional DoA estimation algorithms may not be suited for IIoT applications. In this work, a joint DOA and carrier frequency offset (CFO) schemes is introduced for typical uplink IIoT systems by equipping the receiver with a leaky-wave antenna (LWA). By considering that the LWA can simultaneously steer its beam direction towards the active user. As a result, each user is estimated independently. The joint algorithm is developed based on the high-resolution MUSIC scheme, which utilizes the LWA properties to compute the DoA spectrum. Furthermore, we develop an iterative CFO estimation approach as the desired solution to a multi-dimensional search. Subsequently, we further formulate the Cramer–Rao lower bounds (CRLB) to serve as a performance benchmark for the accuracy of the proposed scheme. The SDR platform is provided to validate the proposed algorithm in real-time. Finally, the simulation results and the SDR experiments justify that the proposed scheme offers more robustness than conventional counterparts with linear array antenna, which achieves a root mean square error (RMSE) approach to the derived CLRB, especially at a medium signal-to-noise ratio (SNR) range which suited for IIoT applications.

Many direction finding methods have been developed these last decades and the performance of some of the latter have been derived with or without modelling errors, jointly with some Cramer Rao bounds (CRB) on the estimated Direction of Arrival (DOAs). However, despite of these works, the link between the array geometry and these performances, or CRB, has been scarcely analyzed, which prevents from optimizing the array design to obtain a specified level of performance. We consider in this paper the general 2D DOA estimation problem from 3D, 2D or 1D arrays and we limit the analysis to the single source case. In this context, the first purpose of this paper is to show that both the deterministic and the stochastic CRB, jointly with the variance of DOAs obtained from the MUSIC algorithm, with and without modelling errors, are proportional to the same term which depends on the sensors location. This term analysis allows to develop the second purpose of the paper, i.e. the first tools for an array design methodology for the performance optimization of DOA estimation methods.

Tensor decomposition is one of the model reduction techniques for compressing deep neural networks. Existing methods use either Tucker decomposition (TD) or Canonical Polyadic decomposition (CPD) for model compression, but none of them tried to combine those two methods, owing to the complexity of choosing a proper decomposition method for each layer. In this paper, we adopted the automatic tuning technique to design an algorithm that can mix both tensor decomposition methods, called Mixed Tensor Decomposition (MTD). The goal is to achieve better compression ratio while keeping similar accuracy as the original models. We used VGG type networks for the case study since they are relatively heavy and computationally expensive. We first studied the relation of model accuracy and compression ratio for Tucker and CPD applying to convolution neural networks (CNN). Based on the studied results, we designed a strategy to select the most suitable decomposition method for each layer, and further fine-tunes the models to recover the accuracy. We have conducted experiments using VGG11 and VGG16 with CIFAR10 dataset, and compared MTD with other tensor decomposition algorithms. The results show that MTD can achieve compression ratio 32 × and 37 × for VGG11 and VGG16 respectively with less than 1% accuracy drops, which is much better than the state-of-the-art tensor decomposition algorithms for model compression.

The separation of Electroencephalography (EEG) sources is a typical application of tensor decompositions in biomedical engineering. The objective of most approaches studied in the literature consists in providing separate spatial maps and time signatures for the identified sources. However, for some applications, a precise localization of each source is required. To achieve this, a two-step approach has been proposed. The idea of this approach is to separate the sources using the canonical polyadic decomposition in the first step and to employ the results of the tensor decomposition to estimate distributed sources in the second step, using the so-called disk algorithm. In this paper, we propose to combine the tensor decomposition and the source localization in a single step. To this end, we directly impose structural constraints, which are based on a priori information on the possible source locations, on the factor matrix of spatial characteristics. The resulting optimization problem is solved using the alternating direction method of multipliers, which is incorporated in the alternating least squares tensor decomposition algorithm. Realistic simulations with epileptic EEG data confirm that the proposed single-step source localization approach outperforms the previously developed two-step approach.

Two-dimensional multiple signal classification (MUSIC) algorithm based on polarization sensitive array (PSA) has excellent performance. However, it suffers a high computational complexity due to a multitude of complex operations. In this paper, we propose a real-valued two-dimensional MUSIC algorithm based on conjugate centrosymmetric signal model, which is applicable to arbitrary centrosymmetric polarization sensitive array. The modified forward/backward averaging, which can be applied to the PSA, is presented. Hence, the eigen-decomposition analysis process and spectrum function computation are converted into real domain, prominently reducing the computational complexity. Then, the direction-of-arrival (DOA) estimation is decoupled from the polarization parameter estimation so that the four-dimensional spectral peak search process is avoided. The theoretical computational complexity is discussed and the Cramer-Rao bound (CRB) of DOA estimation is derived in this paper. The simulation results indicate that the proposed algorithm achieves superior accuracy in DOA estimation and has low computational complexity.

Functional near-infrared spectroscopy (fNIRS) provides an effective tool in neuroscience studies of cognition in infants. fNIRS signals are normally processed by applying ANOVA analysis on the grand average of the hemodynamic responses to investigate the cognitive-related differences between experimental groups. However, this averaging approach does not account for any differences in the temporal patterns of the responses. Therefore, we propose a new approach based on a combination of tensor decomposition and ANOVA. First, a four-way tensor of the hemodynamic responses is arranged as time × frequency × channel× subject and decomposed using Canonical Polyadic Decomposition (CPD). Next, ANOVA is applied to identify significant patterns between subjects. Instead of averaging, the CPD can capture the distinct patterns between groups in all the dimensions. We used fNIRS dataset of 70 infants who participated in an experiment to investigate cortical activation to an agent (i.e., mechanical claws vs. human hands) with different events (i.e., function and non-function). In the comparison with the traditional ANOVA, CPD+ANOVA identified the same significance factors. However, CPD+ANOVA discovered new information on the temporal and spatial patterns indicating a longer interval hemodynamic responses, which was missed using the traditional ANOVA. This new analysis of hemodynamic responses as captured using fNIRS will improve neuroscience and cognitive studies.