Joint Eigenvalue Decomposition Research Articles

The joint eigenvalue decomposition algorithm based on first-order Taylor expansion via the exterior penalty function method

The Joint Eigenvalue Decomposition (JEVD) problem has many applications, especially in the field of signal processing. In this paper, we establish two new algorithms to solve the JEVD for the real or complex case. As the JEVD algorithm requires to inverse a matrix, it can be usefully derived by using a first order Taylor expansion under a reasonable condition.In order to possibly enforce the condition, we propose to take advantage of the Penalty Function Method. By adding a quadratic penalty to the regularized criterion, we turn the constrained problem into a single unconstrained problem. Consequently, we show that this regularized criterion leads to a regularized optimal solution. Thus, we propose the Joint Eigenvalue Decomposition Algorithm Based on First-order Taylor Expansion via the Exterior Penalty Function Method (JDTE-EP) and Sweeping JDTE-EP (SJDTE-EP) algorithms based on this idea. And the two algorithms turn out to enjoy low complexity and good accuracy in all numerical experiments.

Generalized Givens Rotations Applied to Complex Joint Eigenvalue Decomposition

This paper shows the different ways of using generalized Givens rotations in complex joint eigenvaluedecomposition (JEVD) problem. It presents the different schemes of generalized Givens rotation, justifies the introducedapproximations and focuses on the process of extending an algorithm developed for real JEVD to the complex JEVD.Several Joint Diagonalization problem use generalized Givens rotations to achieve the solution, many algorithmsdeveloped in the real case exist in the literature and are not generalized to the complex case. Hence, we show herein asimple and not trivial way to get the complex case from the real one. Simulation results are provided to highlight theeffectiveness and behaviour of the proposed techniques for different scenarios.

A two-step algorithm for joint EigenValue decomposition - Application to canonical polyadic decomposition of fluorescence spectra

In this paper, we propose a new Joint EigenValue Decomposition (JEVD) algorithm. JEVD problem belongs to the family of joint diagonalization problems. Hence, JEVD algorithms aim at estimating the common basis of eigenvectors of a matrix set. This problem occurs in many signal processing applications. It has notably allowed to develop efficient algorithms for the Canonical Polyadic Decomposition (CPD) of multiway arrays. The proposed JEVD algorithm is based on an original two-step approach. The first step consists in transforming the considered matrix set into a set of positive definite matrices. In this purpose, we introduce an ad hoc joint symmetrization algorithm. This first step allows us to transform the JEVD problem into a simpler orthogonal joint diagonalization problem. The second step is then performed using an efficient orthogonal joint diagonalization algorithm of the literature. Eventually, the performance of the proposed approach is deeply investigated in the CPD context of multidimensional fluorescence data. More particularly, we consider difficult scenarios such as the cases of an overestimated rank and highly correlated factors.

Three-Dimensional High-Resolution MIMO Radar Imaging via OFDM Modulation and Unitary ESPRIT

Imaging and recognition of targets with complex maneuvers bring a new challenge to conventional radar applications. In this paper, the three-dimensional (3D) high-resolution image is attained in real-time by a Multiple-Input-Multiple-Output (MIMO) radar system with single Orthogonal-Frequency-Division-Multiplexing (OFDM) pulse. First, to build the orthogonal transmit waveform set for MIMO transmission, we utilize complex orthogonal designs (CODs) for OFDM subcarrier modulation. Based on the OFDM modulation, a preprocessing method is developed for transmit waveform separation without conventional matched filtering. The result array manifold is the Kronecker product of the steering vectors of subcarrier/transmit antenna/receive antenna uniform linear arrays (ULAs). Then, the high-resolution image of target is attained by the Multidimensional Unitary Estimation of Signal Parameters via Rotational Invariant Techniques (MD-UESPRIT) algorithm. The proposed imaging procedures include the multidimensional spatial smoothing, the unitary transform via backward-forward averaging, and the joint eigenvalue decomposition (JEVD) algorithm for automatically paired coordinates estimation. Simulation tests compare the reconstruction results with the conventional methods and analyze the estimation precision relative to signal-to-noise ratio (SNR), system parameters, and errors.

Joint EigenValue Decomposition Algorithms Based on First-Order Taylor Expansion

In this paper, we propose a new approach to compute the Joint EigenValue Decomposition (JEVD) of real or complex matrix sets. JEVD aims to find a common basis of eigenvectors to a set of matrices. JEVD problem is encountered in many signal processing applications. In particular, recent and efficient algorithms for the Canonical Polyadic Decomposition (CPD) of multiway arrays resort to a JEVD step. The suggested method is based on multiplicative updates. It is distinguishable by the use of a first-order Taylor Expansion to compute the inverse of the updating matrix. We call this approach Joint eigenvalue Decomposition based on Taylor Expansion (JDTE). This approach is derived in two versions based on simultaneous and sequential optimization schemes respectively. Here, simultaneous optimization means that all entries of the updating matrix are simultaneously optimized at each iteration. To the best of our knowledge, such an optimization scheme had never been proposed to solve the JEVD problem in a multiplicative update procedure. Our numerical simulations show that, in many situations involving complex matrices, the proposed approach improves the eigenvectors estimation while keeping a limited computational cost. Finally, these features are highlighted in a practical context of source separation through the CPD of telecommunication signals.

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.

Joint Eigenvalue Decomposition of Non-Defective Matrices Based on the LU Factorization With Application to ICA

In this paper we propose a fast and efficient Jacobi-like approach named JET (Joint Eigenvalue decomposition based on Triangular matrices) for the Joint EigenValue Decomposition (JEVD) of a set of real or complex non-defective matrices based on the LU factorization of the matrix of eigenvectors. Contrarily to classical Jacobi-like JEVD methods, the iterative procedure of the JET approach can be reduced to the search for only one of the two triangular matrices involved in the factorization of the matrix of eigenvectors, hence decreasing the numerical complexity. Two variants of the JET technique, namely JET-U and JET-O, which correspond to the optimization of two different cost functions are described in detail and these are extended to the complex case. Numerical simulations show that in many practical cases the JET approach provides more accurate estimation of the matrix of eigenvectors than its competitors and that the lowest numerical complexity is consistently achieved by the JET-U algorithm. In addition, we illustrate in the ICA context the interest of being able to solve efficiently the (non-orthogonal) JEVD problem. More particularly, based on our JET-U algorithm, we propose a more robust version of an existing ICA method, named MICAR-U. The identifiability of the latter is studied and proved under some conditions. Computer results given in the context of brain interfaces show the better ability of MICAR-U to denoise simulated electrocortical data compared to classical ICA techniques for low signal to noise ratio values.

Canonical Polyadic Decomposition based on joint eigenvalue decomposition

A direct algorithm based on Joint EigenValue Decomposition (JEVD) has been proposed to compute the Canonical Polyadic Decomposition (CPD) of multi-way arrays (tensors). The iterative part of our method is thus limited to the JEVD computation. At this occasion we also propose an original JEVD technique. Most of the iterative CPD algorithms such as ALS have been shown by means of practical studies to suffer from convergence problems (local minima, slow convergence or high computational cost per iteration). On the other hand, direct methods seem in practice to confine these disadvantages but impose some restrictive necessary conditions. In this context, our proposed algorithm involves less restrictive necessary conditions than other recent direct approaches and a limited computational complexity. It has been compared to reference (direct and non-direct) algorithms on synthetic arrays and real spectroscopic data. These numerical examples highlight the main advantages of the proposed methods to solve both the JEVD and CPD problems.