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Sensitivity analysis for the problem of matrix joint diagonalization,” to appear at
 SIAM journal on matrix analysis and applications (Special Issue on Tensorial Methods
, 2008
"... Abstract. We investigate the sensitivity of the problem of NonOrthogonal (matrix) Joint Diagonalization (NOJD). First, we consider the uniqueness conditions for the problem of Exact Joint Diagonalization (EJD), which is closely related to the issue of uniqueness in tensor decompositions. As a bypr ..."
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Abstract. We investigate the sensitivity of the problem of NonOrthogonal (matrix) Joint Diagonalization (NOJD). First, we consider the uniqueness conditions for the problem of Exact Joint Diagonalization (EJD), which is closely related to the issue of uniqueness in tensor decompositions. As a byproduct, we derive the wellknown identifiability conditions for Independent Component Analysis (ICA), based on an EJD formulation of ICA. We introduce some known cost functions for NOJD and derive flows based on these cost functions for NOJD. Then we define and investigate the noise sensitivity of the stationary points of these flows. We show that the condition number of the joint diagonalizer and uniqueness of the joint diagonalizer as measured by modulus of uniqueness (as defined in the paper) affect the sensitivity. We also investigate the effect of the number of matrices on the sensitivity. Our numerical experiments confirm the theoretical results. 1
Blind component separation in wavelet space: Application to CMB analysis
 EURASIP Journal on Applied Signal Processing
, 2005
"... It is a recurrent issue in astronomical data analysis that observations are incomplete maps with missing patches or intentionally masked parts. In addition, many astrophysical emissions are non stationary processes over the sky. All these effects impair data processing techniques which work in the F ..."
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It is a recurrent issue in astronomical data analysis that observations are incomplete maps with missing patches or intentionally masked parts. In addition, many astrophysical emissions are non stationary processes over the sky. All these effects impair data processing techniques which work in the Fourier domain. Spectral matching ICA (SMICA) is a source separation method based on spectral matching in Fourier space designed for the separation of diffuse astrophysical emissions in Cosmic Microwave Background observations. This paper proposes an extension of SMICA to the wavelet domain and demonstrates the effectiveness of waveletbased statistics for dealing with gaps in the data.
Penalty functionbased joint diagonalization approach for convolutive blind separation of nonstationary sources
 IEEE Transactions on Signal Processing
, 2005
"... Abstract—A new approach for convolutive blind source separation (BSS) by explicitly exploiting the secondorder nonstationarity of signals and operating in the frequency domain is proposed. The algorithm accommodates a penalty function within the crosspower spectrumbased cost function and thereby ..."
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Abstract—A new approach for convolutive blind source separation (BSS) by explicitly exploiting the secondorder nonstationarity of signals and operating in the frequency domain is proposed. The algorithm accommodates a penalty function within the crosspower spectrumbased cost function and thereby converts the separation problem into a joint diagonalization problem with unconstrained optimization. This leads to a new member of the family of joint diagonalization criteria and a modification of the search direction of the gradientbased descent algorithm. Using this approach, not only can the degenerate solution induced by a null unmixing matrix and the effect of large errors within the elements of covariance matrices at lowfrequency bins be automatically removed, but in addition, a unifying view to joint diagonalization with unitary or nonunitary constraint is provided. Numerical experiments are presented to verify the performance of the new method, which show that a suitable penalty function may lead the algorithm to a faster convergence and a better performance for the separation of convolved speech signals, in particular, in terms of shape preservation and amplitude ambiguity reduction, as compared with the conventional secondorder based algorithms for convolutive mixtures that exploit signal nonstationarity. Index Terms—Blind source separation, convolutive mixtures, frequency domain, orthogonal/nonorthogonal constraints, penalty function, speech signals. I.
Blind Source Separation and Independent Component Analysis: A Review
, 2004
"... Blind source separation (BSS) and independent component analysis (ICA) are generally based on a wide class of unsupervised learning algorithms and they found potential applications in many areas from engineering to neuroscience. A recent trend in BSS is to consider problems in the framework of matr ..."
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Blind source separation (BSS) and independent component analysis (ICA) are generally based on a wide class of unsupervised learning algorithms and they found potential applications in many areas from engineering to neuroscience. A recent trend in BSS is to consider problems in the framework of matrix factorization or more general signals decomposition with probabilistic generative and tree structured graphical models and exploit a priori knowledge about true nature and structure of latent (hidden) variables or sources such as spatiotemporal decorrelation, statistical independence, sparseness, smoothness or lowest complexity in the sense e.g., of best predictability. The possible goal of such decomposition can be considered as the estimation of sources not necessary statistically independent and parameters of a mixing system or more generally as finding a new reduced or hierarchical and structured representation for the observed (sensor) data that can be interpreted as physically meaningful coding or blind source estimation. The key issue is to find a such transformation or coding (linear or nonlinear) which has true physical meaning and interpretation. We present a review of BSS and ICA, including various algorithms for static and dynamic models and their applications. The paper mainly consists of three parts:
Contrast Functions For Blind Separation And Deconvolution Of Sources
, 2001
"... A general method to construct contrast functions for blind source separation is presented. It is based on a superadditive functional of class II applied to the distributions of the reconstructed sources. Examples of such functionals are given. Our approach permits exploiting the temporal dependence ..."
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A general method to construct contrast functions for blind source separation is presented. It is based on a superadditive functional of class II applied to the distributions of the reconstructed sources. Examples of such functionals are given. Our approach permits exploiting the temporal dependence of the sources by using a functional on the joint distribution of the source process over a time interval. This yields many new examples and frees us from the constraint that the sources be non Gaussian. Contrasts functions based on cumulants requiring the orthogonality constraint is also covered. Finally, the case of convolutive mixtures is considered in relation with the problem of blind separationdeconvolution. 1.
P.S.Krishnaprasad “Some Gradient Based Joint Diagonalization Methods for ICA
 in Proceedings of the 5th International Conference on Independent Component Analysis and Blind Source Separation
"... Abstract. We present a set of gradient based orthogonal and nonorthogonal matrix joint diagonalization algorithms. Our approach is to use the geometry of matrix Lie groups to develop continuoustime flows for joint diagonalization and derive their discretized versions. We employ the developed method ..."
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Abstract. We present a set of gradient based orthogonal and nonorthogonal matrix joint diagonalization algorithms. Our approach is to use the geometry of matrix Lie groups to develop continuoustime 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 nonorthogonal joint diagonalization. These algorithms prewhiten or sphere the data but do not restrict the subsequent search for the (reduced) unmixing matrix to orthogonal matrices, hence they make effective use of both second and higher order statistics. 1
Tensors: a Brief Introduction
, 2014
"... Tensor decompositions are at the core of many Blind Source Separation (BSS) algorithms, either explicitly or implicitly. In particular, the Canonical Polyadic (CP) tensor ..."
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Tensor decompositions are at the core of many Blind Source Separation (BSS) algorithms, either explicitly or implicitly. In particular, the Canonical Polyadic (CP) tensor
A Jacobitype method for computing orthogonal tensor decompositions
 SIAM J. Matrix Anal. Appl
, 2006
"... Abstract. Suppose A =(aijk) ∈ Rn×n×n is a threeway array or thirdorder tensor. Many of the powerful tools of linear algebra such as the singular value decomposition (SVD) do not, unfortunately, extend in a straightforward way to tensors of order three or higher. In the twodimensional case, the SV ..."
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Abstract. Suppose A =(aijk) ∈ Rn×n×n is a threeway array or thirdorder tensor. Many of the powerful tools of linear algebra such as the singular value decomposition (SVD) do not, unfortunately, extend in a straightforward way to tensors of order three or higher. In the twodimensional case, the SVD is particularly illuminating, since it reduces a matrix to diagonal form. Although it is not possible in general to diagonalize a tensor (i.e., aijk = 0 unless i = j = k), our goal is to “condense ” a tensor in fewer nonzero entries using orthogonal transformations. We propose an algorithm for tensors of the form A∈Rn×n×n that is an extension of the Jacobi SVD algorithm for matrices. The resulting tensor decomposition reduces A to a form such that the quantity ∑n i=1 a2 iii or ∑n i=1 aiii is maximized. Key words. multilinear algebra, tensor decomposition, singular value decomposition, multidimensional arrays
Exploiting source nonstationary and coloration in blind source separation
 In Proc. Int. Conf. on Digital Signal Processing (DSP2002
, 2002
"... A new method for blind sources separation of instantaneous mixtures is developed. It exploits both the spectral and time and diversity of the sources and is based on the Gaussian mutual information. As a result, it uses only second order statistics and can be efficiently implemented through a joint ..."
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A new method for blind sources separation of instantaneous mixtures is developed. It exploits both the spectral and time and diversity of the sources and is based on the Gaussian mutual information. As a result, it uses only second order statistics and can be efficiently implemented through a joint diagonalization algorithm. Simulation result illustrates the good performance of the method. 1.
Penalty function based joint diagonalization approach for convolutive blind separation of nonstationary sources
 IEEE Trans. Sig. Proc
, 2004
"... In this paper, we address convolutive blind source separation (BSS) of speech signals in the frequency domain and explicitly exploit the second order statistics (SOS) of nonstationary signals. Based on certain constraints on the BSS solution, we propose to reformulate the problem as an unconstrain ..."
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Cited by 8 (6 self)
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In this paper, we address convolutive blind source separation (BSS) of speech signals in the frequency domain and explicitly exploit the second order statistics (SOS) of nonstationary signals. Based on certain constraints on the BSS solution, we propose to reformulate the problem as an unconstrained optimization problem by using nonlinear programming techniques. The proposed algorithm therefore utilizes penalty functions with the crosspower spectrum based criterion and thereby converts the task into a joint diagonalization problem with unconstrained optimization. Using this approach, not only can the degenerate solution induced by a null unmixing matrix and the overlearning effect existing at low frequency bins be automatically removed, but a unifying view to joint diagonalization with unitary or nonunitary constraint is provided. Numerical experiments verify the validity of the proposed approach. 1.