Multilinear Control Systems Theory

Chen, Can; Surana, Amit; Bloch, Anthony M.; Rajapakse, Indika

doi:10.1137/19m1262589

Cited by 23 publications

(31 citation statements)

References 30 publications

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“…• The determinant of a tensor is defined as the product of its eigenvalues i.e., det(A) = i1,...,i N D i1,i2,...,i N ,i1,i2,...,i N . It is also sometimes referred as the unfolding determinant [20], [37].…”

Section: B Basic Definitionsmentioning

confidence: 99%

“…Hence matrices and vectors can be seen as order 2 and order 1 tensors respectively [2]. Tensors have found widespread applications in various engineering disciplines including computer vision [3], [4], signal processing [5]- [8], big data and machine learning [9]- [12], image processing [13], [14], communications [15]- [18], and multi-linear system theory [19], [20]. Our work considers the MMSE estimation problem in context of tensors, thereby extending the basic subject beyond the common vectors and matrices settings.…”

Section: Introductionmentioning

confidence: 99%

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A Tensor Framework for Multi-Linear Complex MMSE Estimation

Pandey

Leib

2021

IEEE Open J. Signal Process.

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Tensors are higher order generalization of vectors and matrices which can be used to describe signals indexed by more than two indices. This paper introduces a tensor framework for minimum mean square error (MMSE) estimation for multi-domain signals and data using the Einstein Product. The framework addresses both proper and improper complex tensors. The multi-domain nature of tensors has been harnessed to provide an augmented representation of improper complex tensors to account for covariance and pseudo-covariance. The classical notions of linear and widely linear MMSE estimators are extended to tensor case leading to the notion of multi-linear and widely multi-linear MMSE estimation. The Tucker product based n-mode Wiener filtering approach more commonly used in tensor estimation has been shown to be a special case of the proposed multi-linear MMSE estimation. An application of the tensor based estimation in a multiple antenna Orthogonal Frequency Division Multiplexing (MIMO OFDM) system is presented where the tensor formulation allows a convenient treatment of inter-carrier interference. A comparison between the tensor estimation and per sub-carrier estimation used for MIMO OFDM is presented which shows a significant performance advantage of using the tensor framework.

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Section: B Basic Definitionsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

A Tensor Framework for Multi-Linear Complex MMSE Estimation

Pandey

Leib

2021

IEEE Open J. Signal Process.

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“…A detailed treatment of such tensor algebra results can be found in [9,[15][16][17][19][20][21]. The Einstein product can be effectively used for representing multi-linear systems of equations, and has been recently employed to develop the notion of multi-linear system theory [18,22]. In the following section, we use the Einstein product to develop a system model for a multi-domain communication system.…”

Section: Notationmentioning

confidence: 99%

“…Using the properties of the Einstein product, several well-known linear algebra relations can be extended to a multi-linear setting without the requirement of any tensor to vector/matrix transformation. The multi-linear algebra notions developed using the Einstein product preserves the natural tensor structure of the associated quantities and therefore has various applications in engineering disciplines [10,[15][16][17][18]. In our work, we consider a multi-linear tensor framework that reveals the latent trade-off that exists between multiple domains.…”

Section: Introductionmentioning

confidence: 99%

Multi-Domain Communication Systems and Networks: A Tensor-Based Approach

Pandey

Venugopal

Leib

2021

Network

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Most modern communication systems, such as those intended for deployment in IoT applications or 5G and beyond networks, utilize multiple domains for transmission and reception at the physical layer. Depending on the application, these domains can include space, time, frequency, users, code sequences, and transmission media, to name a few. As such, the design criteria of future communication systems must be cognizant of the opportunities and the challenges that exist in exploiting the multi-domain nature of the signals and systems involved for information transmission. Focussing on the Physical Layer, this paper presents a novel mathematical framework using tensors, to represent, design, and analyze multi-domain systems. Various domains can be integrated into the transceiver design scheme using tensors. Tools from multi-linear algebra can be used to develop simultaneous signal processing techniques across all the domains. In particular, we present tensor partial response signaling (TPRS) which allows the introduction of controlled interference within elements of a domain and also across domains. We develop the TPRS system using the tensor contracted convolution to generate a multi-domain signal with desired spectral and cross-spectral properties across domains. In addition, by studying the information theoretic properties of the multi-domain tensor channel, we present the trade-off between different domains that can be harnessed using this framework. Numerical examples for capacity and mean square error are presented to highlight the domain trade-off revealed by the tensor formulation. Furthermore, an application of the tensor framework to MIMO Generalized Frequency Division Multiplexing (GFDM) is also presented.

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“…Tensor decomposition techniques such as CANDECOMP/PARAFAC decomposition, 24,25 higher‐order singular value decomposition, 26,27 tensor train decomposition 28,29 help reveal such hidden patterns/redundancies to obtain a compact representation, reducing storage efforts, and enabling efficient computations. Besides, tensor algebra has been recently exploited in systems and control applications 30 . For example, the improved fault diagnosis algorithm‐based the tensor decomposition methods for the complex discrete‐time systems, 31 the tensor network Kalman filter for the recursive identification of high‐order discrete time nonlinear MIMO Volterra systems, 32 the efficient solution of Lyapunov equations based on the quantized tensor train numerical linear algebra 33 …”

Section: Introductionmentioning

confidence: 99%

An efficient recursive identification algorithm for multilinear systems based on tensor decomposition

Wang

Yang

2021

Intl J Robust & Nonlinear

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There are many important fields involving the multilinear system identification. A great number of parameters to be identified is an important challenge, leading to the need for tensorial decomposition and modeling of such systems. This article is about the parameter estimation of the higher‐order multilinear systems with non‐Gaussian noises and to explore the role of tensor algebra in the multilinear model identification. A high‐dimension system identification problem is reformulated in terms of low‐dimension problems by using the tensorial decomposition technique. Further, applying the multi‐innovation identification theory, the recursive algorithm combining with the logarithmic p‐norms is investigated for multilinear systems with non‐Gaussian noises of low computational complexity. Finally, some simulation results illustrate the effectiveness of the proposed recursive identification method.

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Multilinear Control Systems Theory

Cited by 23 publications

References 30 publications

A Tensor Framework for Multi-Linear Complex MMSE Estimation

A Tensor Framework for Multi-Linear Complex MMSE Estimation

Multi-Domain Communication Systems and Networks: A Tensor-Based Approach

An efficient recursive identification algorithm for multilinear systems based on tensor decomposition

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