A Nonorthogonal Approximate Joint Diagonalization Method Excluding Degenerate Solution for Blind Source Separation

Non-orthogonal joint diagonalization algorithm preventable ill conditioned solutions for blind source separation

In this paper, we propose a class of complex non-orthogonal joint diagonalization (NOJD) algorithms with successive rotations. The proposed methods consider LU or LQ decompositions of the mixing matrices, and propose to solve the NOJD problem via two successive stages: L-stage and U (or Q)-stage. Moreover, as the manifolds of target matrices in these stages could be appropriately parameterized by a sequence of simple elementary triangular or unitary matrices, which depend on only one or two parameters, the high-dimensional minimization problems could be replaced by a sequence of lower-dimensional ones. As such, the proposed algorithms are of simple closed-form in each iteration, and do not require the target matrices to be Hermitian nor positive definite. Simulations are provided to compare the proposed methods to other complex NOJD methods.KeywordsComplex non-orthogonal joint diagonalizationBlind source separationLULQ

The paper proposes a new BSS algorithm based on the second-order statistics. By jointly diagonalizing the time delay correlation matrix of the observed signals and using the improved new non-orthogonal joint diagonalization (NOJD) method, a better solution is achieved. The proposed algorithm can successfully separate communication signals under SNR as low as 10dB and the over-determined mode regardless of the signals’ modulation methods. Signal to Interference Noise Ratio (SINR) is used to prove the superiority of the proposed algorithm over the classical Second-Order Blind Identification (SOBI).

We present a set of gradient based orthogonal and non-orthogonal matrix joint diagonalization algorithms. Our approach is to use the geometry of matrix Lie groups to develop continuous-time flows for joint diagonalization and derive their discretized versions. We employ the developed methods to construct a class of Independent Component Analysis (ICA) algorithms based on non-orthogonal joint diagonalization. These algorithms pre-whiten or sphere the data but do not restrict the subsequent search for the (reduced) un-mixing matrix to orthogonal matrices, hence they make effective use of both second and higher order statistics.KeywordsIndependent Component AnalysisIndependent Component AnalysisBlind Source SeparationHigh Order StatisticIndependent Component Analysis AlgorithmThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Joint estimation of direction-of-arrival (DOA) and polarization with electromagnetic vector-sensors (EMVS) is considered in the framework of complex-valued non-orthogonal joint diagonalization (CNJD). Two new CNJD algorithms are presented, which propose to tackle the high dimensional optimization problem in CNJD via a sequence of simple sub-optimization problems, by using LU or LQ decompositions of the target matrices as well as the Jacobi-type scheme. Furthermore, based on the above CNJD algorithms we present a novel strategy to exploit the multi-dimensional structure present in the second-order statistics of EMVS outputs for simultaneous DOA and polarization estimation. Simulations are provided to compare the proposed strategy with existing tensorial or joint diagonalization based methods.

Approximate joint block diagonalization (AJBD) of a set of matrices has applications in blind source separation, e.g., when the signal mixtures contain mutually independent subspaces of dimension higher than one. The main message of this paper is that certain ordinary approximate joint diagonalization (AJD) methods (which were originally derived for “degenerate” subspaces of dimension 1) can also be used successfully for AJBD, but not all are suitable equally well. In particular, we prove that when the set is exactly jointly block-diagonalizable, perfect block-diagonalization is attainable by the recently proposed AJD algorithm “U-WEDGE” (uniformly weighted exhaustive diagonalization with Gaussian iteration) - but this basic consistency property is not shared by some other popular AJD algorithms. In addition, we show using simulation, that in the more general noisy case, the subspace identification accuracy of U-WEDGE compares favorably to competitors.

A fast algorithm, named Complex-Valued Fast Frobenius DIAGonalization (CVFFDIAG), is proposed for seeking the nonunitary approximate joint diagonalizer of a given set of complex-valued target matrices. It adopts a multiplicative update to minimize the Frobenius-norm formulation of the approximate joint diagonalization problem. At each of multiplicative iterations, a strictly diagonally dominant updated matrix is obtained. This scheme ensures the invertibility of the diagonalizer. The CVFFDIAG relaxes several constraints on the target matrices and thus has much general applications. Furthermore, the special approximation of the cost function, the ingenious utilization of some structures and the adequate notation of concerned variables lead to the high computational efficiency of the proposed algorithm. Numerical simulations are conducted to illustrate good performances of the CVFFDIAG.

In this paper, we propose a novel approach to feature compensation performed in the cepstral domain. We apply the linear approximation method in the cepstral domain to simplify the relationship among clean speech, noise and noisy speech. Conventional log-spectral domain feature compensation methods usually assume that each log-spectral coefficient is independent, which is far from real observations. Processing in the cepstral domain has the advantage that the spectral correlation among different frequencies are taken into consideration. By using the diagonal covariance approximation, we can easily modify the conventional log-spectral domain feature compensation technique to fit to the cepstral domain. The proposed approach shows significant improvements in the AURORA2 speech recognition task.

A new approach with piecewise-constant arguments to approximate and numerical solutions of oscillatory problems

A comparative study of approximate joint diagonalization algorithms of a set of matrices is presented. Using a weighted least-squares criterion, without the orthogonality constraint, an algorithm is compared with an analogous one for blind source separation (BSS). The criterion of the present algorithm is on the separating matrix while the other is on the mixing matrix. The convergence of the algorithm is proved under some mild assumptions. The performances of the two algorithms are compared with usual standard algorithms using BSS simulations results. We show that the improvement in estimating the separating matrix, resulting from the relaxation of the orthogonality restriction, can be achieved in presence of additive noise when the length of observed sequences is sufficiently large and when the mixing matrix is not close to an orthogonal matrix

An inverse dynamics method is introduced for constructing exact special-case solutions for hybrid coordinate ordinary/partial systems of differential equations (hybrid ODE/PDE systems). The solution is constructed such that it lies near a given approximate numerical solution, and therefore the special-case solutions can be generated in a versatile and physically meaningful fashion and can serve as a benchmark problem to validate approximate solution methods. The exact solution is constructed such that it is a differentiable, continuous-function neighbor of the given approximate numerical solution. This continuous solution is then substituted into the governing system of ODEs/PDEs and a full complement of distributed and boundary forces is determined algebraically to exactly satisfy the differential equations. This process has been automated by computer symbol manipulation. Since the exact special-case algebraic solutions can be evaluated anywhere in space and time, this approach is ideally suited to providing a true exact motion and the corresponding forces for studying the convergence errors in a family of approximate solutions. This methodology makes it possible for one to rigorously determine exact solution errors for a significant class of ODE/PDE systems for which the initial-value problem is not, in general, exactly solvable. We explore the utility of this method in validating numerical solution methods.

Many optical and other nonlinear media are governed by dispersive, or diffractive, wave equations, for which initial jump discontinuities are resolved into a dispersive shock wave. The dispersive shock wave smooths the initial discontinuity and is a modulated wavetrain consisting of solitary waves at its leading edge and linear waves at its trailing edge. For integrable equations the dispersive shock wave solution can be found using Whitham modulation theory. For nonlinear wave equations which are hyperbolic outside the dispersive shock region, the amplitudes of the solitary waves at the leading edge and the linear waves at the trailing edge of the dispersive shock can be determined. In this paper an approximate method is presented for calculating the amplitude of the lead solitary waves of a dispersive shock for general nonlinear wave equations, even if these equations are not hyperbolic in the dispersionless limit. The approximate method is validated using known dispersive shock solutions and then applied to calculate approximate dispersive shock solutions for equations governing nonlinear optical media, such as nematic liquid crystals, thermal glasses and colloids. These approximate solutions are compared with numerical results and excellent comparisons are obtained.

Soliton-effect compression and dispersive radiation

We propose a new blind separation approach based on the Generalized Generating Function (GGF) of observations for complex sources by generalizing the definition of generating function. A new core equation is obtained and an approximate joint diagonalization scheme is used to estimate the mixing matrix by diagonalizing the Hessian matrix of the second GGF of the observations. Simulation results show that the GGF approach has superior performance to the existing classical algorithms when the SNR of observations is low and the data block is short.

Equations, which have nonlinear nonmonotonic dependence of one of the coefficients on an unknown function, can describe processes of heat and mass transfer. As a rule, existing approximate methods do not provide solutions with acceptable accuracy. Numerical methods do not involve obtaining an analytical expression for the unknown function and require studying the convergence of the algorithm used. The value of absolute error is uncertain. The authors propose an approximate method for solving such problems based on Westphal comparison theorems. The comparison theorems allow finding upper and lower bounds of the unknown exact solution. A special procedure developed for the stepwise improvement of these bounds provide solutions with a given accuracy. There are only a few problems for equations with nonlinear nonmonotonic coefficients for which the exact solution has been obtained. One of such problems, presented in this article, shows the efficiency of the proposed method. The results prove that the proposed method for obtaining bounds of the solution of a nonlinear nonmonotonic equation of parabolic type can be considered as a new method of the approximate analytical solution having guaranteed accuracy. In addition, the proposed here method allows calculating the maximum deviation from the unknown exact solution of the results of other approximate and numerical methods.