Tensor Decompositions and Practical Applications: A Hands-on Tutorial

Tensors are multi-way arrays which can be used to model systems spanning many domains. This work proposes to use tensors for characterizing, analyzing, and designing multi-domain communication systems. Most modern day communication systems make use of coding and modulation across different domains such as space, frequency, time. Hence a unified mathematical framework characterizing such a multiple-domain system in an intuitive manner is well needed. In this paper, we present such a unified framework that characterizes a communication system with N input domains, M output domains and an M + N domains multi-linear tensor channel. The proposed framework is generic where the physical interpretation of the domains is system specific. We illustrate a few examples from multi-antenna multicarrier and multi-user systems that fit the proposed framework. Assuming a fixed tensor channel, we provide an information theoretic analysis by deriving its Shannon capacity and input power allocation under a variety of power constraints. In this paper we show how the tensor framework's suitability to mathematically describe a family of power constraints can be used to design and analyze various multiple domain communication systems. The tensor based approach extends water-filling from a matrix setting to tensors, encapsulating the effects of multiple domains thereby allowing joint multi-domain precoding. We show that the capacity pre-log for a tensor channel increases exponentially in the number of domains, indicating the potential of tensor based multi-domain communication systems to provide the large information transmission rates envisaged for 5G and beyond systems. We also show the application of the tensor framework in characterizing the capacity and rate regions of multi-user MIMO channels. Both multiple access and interference channels are considered where the tensor based approach leads to a coordinated users transmission scheme. Such a scheme ensures higher achievable sum rates as compared to independent user transmissions.INDEX TERMS Tensors, MIMO channels, multi-user MIMO, Shannon capacity, tensor SVD and EVD.

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“…where all the individual vectors x (19) form the columns of the matrix X[k] of size N T × D and all the individual matrices H [1], . .…”

Section: Discrete Tensor System Model a The Tensor Channelmentioning

confidence: 99%

The Tensor Multi-Linear Channel and Its Shannon Capacity

Pandey

Leib

2022

IEEE Access

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“…In this section, we only provide some relevant tensor definitions and properties needed for this paper. More details on tensor algebra can be found in [2], [12], [15], [29], [37], [38]. Definition 1.…”

Section: B Basic Definitionsmentioning

confidence: 99%

“…Notice however, that X X X and Y Y Y being jointly proper is not a necessary condition, as even if X X X is not proper, i.e.C X X X = 0 T butC Y Y Y = 0 T andC XY XY XY = 0 T , still ( 56) is satisfied and multi-linear and widely multi-linear estimate will be same. Further, if the tensor to be estimated X X X is real, the cross pseudo-covariance is given as : 61) into (37) and comparing with (38) shows that A 2 = A * 1 . Thus for real X X X, the widely multi-linear MMSE estimate and the associated mean square error (from ( 44)) are given as :…”

Section: Comparing Multi-linear and Widely Multi-linear Mmse Estimationmentioning

confidence: 99%

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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“…Using properties of the Einstein product, many linear algebra concepts such as inversion, identity, hermitian, trace, determinant, rank, and eigenvalue decomposition (EVD) can be extended to a multi-linear setting. 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].…”

Section: Notationmentioning

confidence: 99%

“…green nodes (20) The summation over indices i 1 , i 2 , i 3 in (20) is reflected in the edges connecting the green and the orange nodes in Figure 5. Specific algorithms to compute the Einstein product, which act directly on the cores of the tensor train format are presented in [33].…”

Section: Application Of Tensor Train Decomposition and Tensor Networkmentioning

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 Decompositions and Practical Applications: A Hands-on Tutorial

Cited by 8 publications

References 21 publications

The Tensor Multi-Linear Channel and Its Shannon Capacity

The Tensor Multi-Linear Channel and Its Shannon Capacity

A Tensor Framework for Multi-Linear Complex MMSE Estimation

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

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