Xavier Luciani

Blind Identification of Underdetermined Mixtures Based on the Characteristic Function: The Complex Case

Blind identification of underdetermined mixtures can be addressed efficiently by using the second ChAracteristic Function (CAF) of the observations. Our contribution is twofold. First, we propose the use of a Levenberg-Marquardt algorithm, herein called LEMACAF, as an alternative to an Alternating Least Squares algorithm known as ALESCAF, which has been used recently in the case of real mixtures of real sources. Second, we extend the CAF approach to the case of complex sources for which the previous algorithms are not suitable. We show that the complex case involves an appropriate tensor stowage, which is linked to a particular tensor decomposition. An extension of the LEMACAF algorithm, called then proposed to blindly estimate the mixing matrix by exploiting this tensor decomposition. In our simulation results, we first provide performance comparisons between third- and fourth-order versions of ALESCAF and LEMACAF in various situations involving BPSK sources. Then, a performance study of is carried out considering 4-QAM sources. These results show that the proposed algorithm provides satisfying estimations especially in the case of a large underdeterminacy level.

Semi-algebraic canonical decomposition of multi-way arrays and Joint Eigenvalue Decomposition

A semi-algebraic algorithm based on Joint EigenValue Decomposition (JEVD) is proposed to compute the CP decomposition of multi-way arrays. The iterative part of the method is thus limited to the JEVD computation. In addition it involves less restrictive hypothesis than other recent semi-algebraic approaches. We also propose an original JEVD technique based on the LU factorization. Numerical examples highlight the main advantages of the proposed methods to solve both the JEVD and CP problems.

CONFAC Decomposition Approach to Blind Identification of Underdetermined Mixtures Based on Generating Function Derivatives

This work proposes a new tensor-based approach to solve the problem of blind identification of underdetermined mixtures of complex-valued sources exploiting the cumulant generating function (CGF) of the observations. We show that a collection of second-order derivatives of the CGF of the observations can be stored in a third-order tensor following a constrained factor (CONFAC) decomposition with known constrained structure. In order to increase the diversity, we combine three derivative types into an extended third-order CONFAC decomposition. A detailed uniqueness study of this decomposition is provided, from which easy-to-check sufficient conditions ensuring the essential uniqueness of the mixing matrix are obtained. From an algorithmic viewpoint, we develop a CONFAC-based enhanced line search (CONFAC-ELS) method to be used with an alternating least squares estimation procedure for accelerated convergence, and also analyze the numerical complexities of two CONFAC-based algorithms (namely, CONFAC-ALS and CONFAC-ELS) in comparison with the Levenberg-Marquardt (LM)-based algorithm recently derived to solve the same problem. Simulation results compare the proposed approach with some higher-order methods. Our results also corroborate the advantages of the CONFAC-based approach over the competing LM-based approach in terms of performance and computational complexity.

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.

Blind identification of underdetermined mixtures based on the hexacovariance and higher-order cyclostationarity

Static linear mixtures with more sources than sensors are considered. Blind identification (BI) of underdetermined mixtures is addressed by taking advantage of sixth order (SixO) statistics and the virtual array (VA) concept. Surprisingly, identification methods solely based on the hexacovariance matrix succeed well, despite their expected high estimation variance; this is due to the inherently good conditioning of the problem. A computationally simple but efficient algorithm, named BIRTH (Blind Identification of mixtures of sources using Redundancies in the daTa Hexacovariance matrix), is proposed and enables the identification of the steering vectors of up to P=N/sup 2/-N+1 sources for arrays of N sensors with space diversity only, and up to P=N/sup 2/ for those with angular and polarization diversities. Five numerical algorithms are compared.

Joint eigenvalue decomposition using polar matrix factorization

In this paper we propose a new algorithm for the joint eigenvalue decomposition of a set of real non-defective matrices. Our approach resorts to a Jacobi-like procedure based on polar matrix decomposition. We introduce a new criterion in this context for the optimization of the hyperbolic matrices, giving birth to an original algorithm called JDTM. This algorithm is described in detail and a comparison study with reference algorithms is performed. Comparison results show that our approach provides quicker and more accurate results in all the considered situations.

A coupled joint eigenvalue decomposition algorithm for canonical polyadic decomposition of tensors

In this paper we propose a novel algorithm to compute the joint eigenvalue decomposition of a set of squares matrices. This problem is at the heart of recent direct canonical polyadic decomposition algorithms. Contrary to the existing approaches the proposed algorithm can deal equally with real or complex-valued matrices without any modifications. The algorithm is based on the algebraic polar decomposition which allows to make the optimization step directly with complex parameters. Furthermore, both factorization matrices are estimated jointly. This “coupled” approach allows us to limit the numerical complexity of the algorithm. We then show with the help of numerical simulations that this approach is suitable for tensors canonical polyadic decomposition.

Deterministic blind separation of sources having different symbol rates using tensor-based parallel deflation

In this work, we address the problem of blind separation of non-synchronous statistically independent sources from underdetermined mixtures. A deterministic tensor-based receiver exploiting symbol rate diversity by means of parallel deflation is proposed. By resorting to bank of samplers at each sensor output, a set of third-order tensors is built, each one associated with a different source symbol period. By applying multiple Canonical Decompositions (CanD) on these tensors, we can obtain parallel estimates of the related sources along with an estimate of the mixture matrix. Numerical results illustrate the bit-error-rate performance of the proposed approach for some system configurations.

A fast algorithm for joint eigenvalue decomposition of real matrices

We introduce an original algorithm to perform the joint eigen value decomposition of a set of real matrices. The proposed algorithm is iterative but does not resort to any sweeping procedure such as classical Jacobi approaches. Instead we use a first order approximation of the inverse of the matrix of eigen vectors and at each iteration the whole matrix of eigenvectors is updated. This algorithm is called Joint eigenvalue Decomposition using Taylor Expansion and has been designed in order to decrease the overall numerical complexity of the procedure (which is a trade off between the number of iterations and the cost of each iteration) while keeping the same level of performances. Numerical comparisons with reference algorithms show that this goal is achieved.

Blind MIMO system identification using constrained factor decomposition of output generating function derivatives

This work addresses the blind identification of complex MIMO systems driven by complex input signals using a new tensor decomposition approach. We show that a collection of successive second-order derivatives of the second generating function of the system outputs can be stored in a higher-order tensor following a constrained factor (CONFAC) decomposition. The proposed decomposition captures the repeated linear combinations involving real and imaginary components of the MIMO system matrix arising from the successive differentiation of outputs generating function derivatives. By exploiting different derivative forms computed at multiple points of the observation space, an “extended” CONFAC decomposition enjoying essential uniqueness is obtained. Thanks to this uniqueness property, a blind estimation of the MIMO system response matrix is possible.