Khatri-Rao Product Research Articles

A characterization of common Lyapunov diagonal stability using Khatri-Rao products

Using the Khatri-Rao product, we presented new characterizations for the common Lyapunov diagonal stability for a family of real matrices $ \mathcal{A} $. For special partitions $ \alpha $, we used the notion of $ \mathcal{P}^{\alpha} $-sets and common $ \alpha $-scalar Lyapunov stability to formulate further characterizations. Furthermore, generalizations of these results to the common $ \alpha $-scalar Lyapunov stability were developed. Our goal of this paper was to unify and enhance relevant work.

Identification of polynomial nonlinear systems based on center manifold

A center-manifold (CM) based system identification (SID) method is developed to identify multi-input-multi-output continuous-time polynomial nonlinear systems in the state–space form. On one hand, the CM method is a natural extension of the frequency-domain subspace method to nonlinear SID problems. It decomposes the nonlinear SID problem into a series of linear least-square problems (in a precise sense), which can be solved analytically. This is in contrast to many existing methods that rely on iterative search of the global minimum. On the other hand, it is also a generalization of the invariant-subspace (ISP) based SID method. It inherits the nice property of the ISP method that the advantages of time-domain and frequency-domain SID methods are combined. In the paper, a persistence of excitation (PE) condition for the CM method is derived, which is described by the rank of certain Khatri–Rao product. Under the PE condition, the identification error is shown to be bounded within an arbitrarily small neighborhood of the origin when measurement noise is presented.

An Enhanced DOA Estimation Method for Coherent Sources via Toeplitz Matrix Reconstruction and Khatri–Rao Subspace

The Toeplitz matrix reconstruction methods are capable of resolving coherent signals, playing a crucial role in the direction-of-arrival (DOA) estimation of acoustic sources. However, the decoherence processing sacrifices the array aperture and further results in a reduced resolution capability for the number of identifiable sources. To solve this issue, we propose an enhanced method using the Khatri–Rao subspace to resolve more coherent sources than that of the existing Toeplitz matrix reconstruction methods. Firstly, a full set of Toeplitz matrices with full rank is obtained. Then, the virtual array aperture can be obtained using the Khatri–Rao product of the array response, and the degree of freedom provided inherently in the virtual array structure is about twice the size of that of the existing Toeplitz methods. Next, linear processing is further used to achieve complexity reduction without losing the effective degree of freedom. Finally, the DOA estimation for more coherent sources can be achieved by combining it with conventional methods. Numerical simulations verify the superiority of the proposed method.

Beamspace-based underdetermined DOA estimator via Khatri-Rao subspace for quasi-stationary signals

The Khatri-Rao subspace methods utilize the characteristics of the Khatri-Rao product of the array response to achieve the extension of the physical aperture and further resolve the quasi-stationary signals in the underdetermined cases. However, the estimation performance of the existing Khatri-Rao subspace methods suffers from the input signal-to-noise (SNR). Especially for a low SNR, the estimation accuracy of the local covariance matrix signals decreases, which will further result in the DOA estimation performance degradation. To address these problems, an enhanced Khatri-Rao subspace method based on beamspace processing for the DOA estimation is proposed. The total array is first split into several identical subarrays, and the beamspace transformation for the data received by subarrays is implemented via beamforming. This results in the improvement of the SNR and further improves the noise suppression ability for the low SNR regions. Then, the Khatri-Rao subspace processing is used for the separated beamspace sources to realize the aperture extension. Finally, the DOAs for the separated sources in the beamspace domain can be further obtained when combined with the subspace-based methods, and the undetermined problem can also be achieved by combining the DOAs obtained from different directions in the beamspace domain.

On Asymptotic Stability of Discrete-Time Hybrid Systems

In this brief, we consider the asymptotic stability problem of a novel class of discrete-time hybrid systems. First, we introduce the so-called system in detail, which consists of discrete-time continuous-valued and Boolean dynamics. Second, we give the algebraic form of the systems by using the Khatri-Rao product and semi-tensor product (i.e., Cheng product). Third, we propose the concept of asymptotic stability for the hybrid systems. Further, based on Lyapunov functions, a necessary and sufficient condition is derived for testing asymptotic stability. Finally, a numerical example is presented to illustrate the effectiveness of the theoretical results.

A Khatri-Rao product tensor network for efficient symmetric MIMO Volterra identification

The identification of symmetric tensor network MIMO Volterra models has been studied earlier via the computation of a Moore-Penrose pseudoinverse in tensor network form. The current state of the art requires the construction of a tensor network of a repeated Khatri-Rao product of a matrix with itself. This construction has a computational complexity that is dominated by one singular value decomposition (SVD) of an RI × IN matrix, where N is the number of measurements, I depends linearly on the number of inputs and input lags and R is the maximal tensor network rank. In this article, we prove an alternative method for constructing this tensor network without any computation whatsoever. The pseudoinverse can then be computed through an orthogonalization of the newly proposed tensor network. Furthermore, the proposed algorithm allows for the recursive identification of symmetric Volterra models of increasing degree D, which reduces the computation to one SVD of a RI × N matrix per step. Through numerical experiments we demonstrate how the proposed algorithm enables up to ten times faster identification of symmetric tensor network MIMO Volterra systems.

On the Support Recovery of Jointly Sparse Gaussian Sources via Sparse Bayesian Learning

In this work, we provide non-asymptotic, probabilistic guarantees for successful recovery of the common nonzero support of jointly sparse Gaussian sources in the multiple measurement vector (MMV) problem. The support recovery problem is formulated as the marginalized maximum likelihood (or type-II ML) estimation of the variance hyperparameters of a joint sparsity inducing Gaussian prior on the source signals. We derive conditions under which the resulting nonconvex constrained optimization perfectly recovers the nonzero support of a joint-sparse Gaussian source ensemble with arbitrarily high probability. The support error probability decays exponentially with the number of MMVs at a rate that depends on the smallest restricted singular value and the nonnegative null space property of the self Khatri-Rao product of the sensing matrix. Our analysis confirms that nonzero supports of size as high as <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$O(m^{2})$ </tex-math></inline-formula> are recoverable from <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$m$ </tex-math></inline-formula> measurements per sparse vector. Our derived sufficient conditions for support consistency of the proposed constrained type-II ML solution also guarantee the support consistency of any global solution of the multiple sparse Bayesian learning (M-SBL) optimization whose nonzero coefficients lie inside a bounded interval. For the case of noiseless measurements, we further show that a single MMV is sufficient for perfect recovery of the <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$k$ </tex-math></inline-formula> -sparse support by M-SBL, provided all subsets of <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$k + 1$ </tex-math></inline-formula> columns of the sensing matrix are linearly independent.

Traffic rate network tomography with higher‐order cumulants

AbstractNetwork tomography aims at estimating source–destination traffic rates from link traffic measurements. This inverse problem was formulated by Vardi in 1996 for Poisson traffic over networks operating under deterministic as well as random routing regimes. In this article, we expand Vardi's second‐order moment matching rate estimation approach to higher‐order cumulant matching with the goal of increasing the column rank of the mapping and consequently improving the rate estimation accuracy. We develop a systematic set of linear cumulant matching equations and express them compactly in terms of the Khatri–Rao product. Both least squares estimation and iterative minimum I‐divergence estimation are considered. We develop an upper bound on the mean squared error (MSE) in least squares rate estimation from empirical cumulants. We demonstrate that supplementing Vardi's approach with the third‐order empirical cumulant reduces its minimum averaged normalized MSE in rate estimation by almost 20% when iterative minimum I‐divergence estimation was used.

Conjugate Augmented Decoupled 3-D Parameters Estimation Method for Near-Field Sources

A near-field source localization method is proposed for 2-D direction-of-arrival (DOA) and range estimation based on a symmetrical cross array. It first employs the conjugate symmetry property of signal autocorrelation for different time delays to construct a conjugate augmented spatial–temporal cross-correlation matrix, then the extended steering vector is decoupled to avoid the usual multidimensional search based on the properties of the Khatri–Rao product, and finally three 1-D MUSIC-type searches are employed to obtain the results. The proposed method can realize automatic pairing of multiple parameters associated with each source and it also works in the underdetermined case. Furthermore, the stochastic Cramer–Rao lower bound with different time delays is derived. Compared with two existing methods, simulation results demonstrate that the proposed method provides satisfactory estimation performance for both the DOA and range parameters at low signal-to-noise ratio and with a small number of snapshots.

On a construction procedure for split-block-plot designs based on the Khatri–Rao product of incidence matrices

Summary The construction of incomplete split-block-plot (SBP) designs for three-factor experiments is investigated. A semi-Kronecker (Khatri–Rao) product of matrices is used in the construction procedure. With this method some generated designs are obtained where column treatments and subplot treatments are allocated in balanced square lattice designs, while row treatments are in a randomized complete block design. Experiments with an orthogonal block structure are considered. In this paper we consider designs which are generally balanced. This allows us to give the stratum efficiency factors for treatment combination contrasts in the proposed split-block-plot designs.

A combined invariant-subspace and subspace identification method for continuous-time state–space models using slowly sampled multi-sine-wave data

The problem of identification of linear continuous-time (CT) state–space (SS) models directly from time-domain data is considered. Given multi-sine waves as excitation signals, this paper develops a new identification method called cISSIM (combined Invariant-Subspace and Subspace Identification Method), which is able to consistently identify CT SS models from slowly-sampled time-domain data in an error-in-variables framework. No prefiltering of the input/output signals is required which distinguishes cISSIM from other existing direct CT identification methods. Moreover, asymptotic unbiased estimates of the system state are obtained. A new persistence of excitation condition is also derived for cISSIM, which is shown to be related with the rank of a certain Khatri–Rao product.

Determinantal inequalities for block Hadamard product and Khatri-Rao product of positive definite matrices

&lt;abstract&gt;&lt;p&gt;In this paper, we first give an alternative proof for a result of Liu et al. in [Math. Inequal. Appl. 20 (2017) 537–542]. Then we present two inequalities for the block Hadamard product and the Khatri-Rao product respectively. The inequalities obtained extend the result of Liu et al.&lt;/p&gt;&lt;/abstract&gt;

Smoothed Lv Distribution Based Three-Dimensional Imaging for Spinning Space Debris

Three-dimensional (3-D) imaging plays a vital role in the recognition of spinning space debris. However, the image may be blurred due to the range migration caused by fast rotation of space debris. Moreover, the image quality, which depends on the estimation accuracy of Doppler frequency and chirp rate of scattering centers, is influenced by cross-terms and sidelobes. In this paper, we propose a novel 3-D imaging method based on smoothed Lv distribution (SLVD). Firstly, the selection criterion for best imaging time based on the time-frequency moment is proposed to guarantee that the echo is approximated as a linear frequency modulation signal. Then, we operate the Khatri-Rao product on the centroid frequency and chirp rate (CFCR) representation and the range-Doppler (RD) image to obtain the 3-D image. To decrease the influence of range migration, we process a short time window during the RD imaging procedure. For cross-term and sidelobe suppression, the SLVD is proposed to obtain the CFCR representation by expressing the Lv distribution (LVD) in a convolution form and introducing a centroid frequency window. Experimental results verify the effectiveness of the proposed imaging method and the good performance of the proposed SLVD for cross-term and sidelobe suppression.

Sparsity-driven deterministic sampling strategy for coded aperture x-ray computed tomography

The sub-sampling strategies in x-ray computed tomography (CT) have gained importance due to their practical relevance. In this direction of research, also known as coded aperture x-ray computed tomography (CAXCT), both random and deterministic strategies have been proposed in the literature. Of the techniques available, those based on compressive sensing (CS) have recently gained more traction because CS-based ideas efficiently exploit inherent duplication present in the system. The quality of the reconstructed CT images nevertheless depends on the sparse signal recovery properties of the sub-sampled Radon matrices. In the present work, we optimize CAXCT deterministically in such a way that the corresponding sub-sampled Radon matrices remain close to the equiangular tight frames for better numerical behaviour. We show that this optimization, involving the Khatri–Rao product, leads to a non-negative sparse approximation. While comparing and contrasting our method with its existing counterparts, we show that the proposed algorithm is computationally less involved. Finally, we demonstrate efficacy of the proposed deterministic sub-sampling strategy in recovering CT images in both noiseless and noisy cases.

Chronological Sailfish Optimizer for Preserving Privacy in Cloud Based on Khatri-Rao Product

Abstract The innovative trends of cloud computing acquired the interest of several individuals or enterprises that started outsourcing data to the cloud servers. Recently, numerous techniques are introduced for facilitating privacy protection on untrusted cloud platforms. However, the classical privacy-preserving techniques failed to prevent leakage and incur huge information loss. This paper introduces the efficient technique, named the chronological sailfish optimizer (CSFO) algorithm for privacy preservation in cloud computing. The proposed CSFO is devised by integrating the chronological concept in SailFish optimizer. The input data are fed to a privacy-preservation process wherein hamming weight-based RSA and Khatri-Rao products are utilized for data privacy. Here, the hamming weighted-based RSA is determined by combining the sha256 algorithm with the hamming weight with Rivest–Shamir–Adleman (HRSA) system. Hence, an optimization-driven algorithm is utilized to evaluate optimal matrix generation to handle both the utility and the sensitive information. Here, the fitness function is newly devised considering, realism, privacy and fitness. The experimentation is performed using four datasets, like Pathway Interaction Database, Hungarian, Cleveland and Switzerland. The proposed CSFO provided superior performance with maximal privacy of 0.2173, maximal realism 0.9456 and maximal fitness of 0.5416.

Low complexity sparse beamspace DOA estimation via single measurement vectors for uniform circular array

In this paper, we present a low complexity sparse beamspace direction-of-arrival (DOA) estimation method for uniform circular array (UCA). In the proposed method, we firstly use the beamspace transformation (BT) to transform the signal model of UCA in element-space domain to that of virtual uniform linear array (ULA) in beamspace domain. Subsequently, by applying the vectoring operator on the virtual ULA-like array signal model, a novel dimension-reduction sparse beamspace signal model is derived based on Khatri-Rao (KR) product, the observation data of which is represented by the single measurement vectors (SMVs) via vectorization of sparse covariance matrix. And then, the DOA estimation is formulated as a convex optimization problem by following the concept of a sparse-signal-representation (SSR) of the SMVs. Finally, simulations are carried out to validate the effectiveness of the proposed method. The results show that without knowledge of the number of signals, the proposed method not only has higher DOA resolution than the subspace-based methods in low signal-to-noise ratio (SNR), but also has far lower computational complexity than other sparse-like DOA estimation methods.

KR product and sparse prior based CNN estimator for 2-D DOA estimation

This paper proposes a method based on Khatri-Rao (KR) product, sparse prior, and convolutional neural networks (CNN) to solve the direction-of-arrival (DOA) estimation problem. Firstly, we use the KR product to expand the degree of freedom (DOF) of the 2-D antenna array. Then we calculate the sparse power spectrum of signals and obtain an RGB image tensor of the spectrum. Finally, we design a CNN group with three different sub-networks to estimate 2-D DOA information. Two of the sub-networks are used for obtaining the spectrum of azimuth angle and elevation angle, respectively. One specific network is designed as the pairing network used for paring azimuth angle with the correct elevation angle. The proposed CNN group is data-driven and does not rely on any prior knowledge of incidence signals. We investigate the feature of estimation error, the root mean squared error (RMSE) responses under different experiment environments, the resolution of the proposed estimation CNN group, and the pairing performance of the proposed pairing network. Comparing with prior estimation methods, the proposed CNN group shows satisfactory estimation accuracy and stability.

True Load Balancing for Matricized Tensor Times Khatri-Rao Product

MTTKRP is the bottleneck operation in algorithms used to compute the CP tensor decomposition. For sparse tensors, utilizing the compressed sparse fibers (CSF) storage format and the CSF-oriented MTTKRP algorithms is important for both memory and computational efficiency on distributed-memory architectures. Existing intelligent tensor partitioning models assume the computational cost of MTTKRP to be proportional to the total number of nonzeros in the tensor. However, this is not the case for the CSF-oriented MTTKRP on distributed-memory architectures. We outline two deficiencies of nonzero-based intelligent partitioning models when CSF-oriented MTTKRP operations are performed locally: failure to encode processors' computational loads and increase in total computation due to fiber fragmentation. We focus on existing fine-grain hypergraph model and propose a novel vertex weighting scheme that enables this model encode correct computational loads of processors. We also propose to augment the fine-grain model by fiber nets for reducing the increase in total computational load via minimizing fiber fragmentation. In this way, the proposed model encodes minimizing the load of the bottleneck processor. Parallel experiments with real-world sparse tensors on up to 1024 processors prove the validity of the outlined deficiencies and demonstrate the merit of our proposed improvements in terms of parallel runtimes.

An Oppenheim type determinantal inequality for the Khatri-Rao product

We first give an Oppenheim type determinantal inequality for the Khatri-Rao product of two block positive semidefinite matrices, and then we extend our result to multiple block matrices. As products, the extensions of Oppenheim type inequalities for the Hadamard product are also included.

SGD_Tucker: A Novel Stochastic Optimization Strategy for Scalable Parallel Sparse Tucker Decomposition

Sparse Tucker Decomposition (STD) algorithms learn a core tensor and a group of factor matrices to obtain an optimal low-rank representation feature for the High-Order, High-Dimension, and Sparse Tensor (HOHDST). However, existing STD algorithms face the problem of intermediate variables explosion which results from the fact that the formation of those variables, i.e., matrices Khatri-Rao product, Kronecker product, and matrix-matrix multiplication, follows the whole elements in sparse tensor. The above problems prevent deep fusion of efficient computation and big data platforms. To overcome the bottleneck, a novel stochastic optimization strategy (SGD Tucker) is proposed for STD which can automatically divide the high-dimension intermediate variables into small batches of intermediate matrices. Specifically, SGD Tucker only follows the randomly selected small samples rather than the whole elements, while maintaining the overall accuracy and convergence rate. In practice, SGD Tucker features the two distinct advancements over the state of the art. First, SGD Tucker can prune the communication overhead for the core tensor in distributed settings. Second, the low data-dependence of SGD Tucker enables fine-grained parallelization, which makes SGD Tucker obtaining lower computational overheads with the same accuracy. Experimental results show that SGD Tucker runs at least 2X faster than the state of the art.