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This paper discusses the estimation and numerical calculation of the probability that the 0-norm and 1-norm solutions of underdetermined linear equations are equivalent in the case of sparse representation. First, we define the sparsity degree of a signal. Two equivalence probability estimates are obtained when the entries of the 0-norm solution have different sparsity degrees. One is for the case in which the basis matrix is given or estimated, and the other is for the case in which the basis matrix is random. However, the computational burden to calculate these probabilities increases exponentially as the number of columns of the basis matrix increases. This computational complexity problem can be avoided through a sampling method. Next, we analyze the sparsity degree of mixtures and establish the relationship between the equivalence probability and the sparsity degree of the mixtures. This relationship can be used to analyze the performance of blind source separation (BSS). Furthermore, we extend the equivalence probability estimates to the small noise case. Finally, we illustrate how to use these theoretical results to guarantee a satisfactory performance in underdetermined BSS. Index Terms--Blind source separation (BSS), equivalence probability, noise, sparse representation, sparsity degree, 0-norm solution, 1-norm solution.

A critical analysis of the canonical correlation analysis (CCA) approach in blind source separation (BSS) is provided. It is proved that by maximizing the autocorrelation functions of the recovered signals we can separate the source signals successfully. It is further shown that the CCA approach represents the same class of generalized eigenvalue decomposition (GEVD) problems as the matrix pencil method. Finally, online realizations of the CCA approach are discussed with a linear-predictor-based algorithm studied as an example. Index Terms--Blind source separation (BSS), canonical correlation analysis (CCA), linear predictor, matrix pencil, second-order statistics (SOS).

In this paper, we use a two-stage sparse factorization approach for blindly estimating the channel parameters and then estimating source components for electroencephalogram (EEG) signals. EEG signals are assumed to be linear mixtures of source components, artifacts, etc. Therefore, a raw EEG data matrix can be factored into the product of two matrices, one of which represents the mixing matrix and the other the source component matrix. Furthermore, the components are sparse in the time-frequency domain, i.e., the factorization is a sparse factorization in the time frequency domain. It is a challenging task to estimate the mixing matrix. Our extensive analysis and computational results, which were based on many sets of EEG data, not only provide firm evidences supporting the above assumption, but also prompt us to propose a new algorithm for estimating the mixing matrix. After the mixing matrix is estimated, the source components are estimated in the time frequency domain using a linear programming method. In an example of the potential applications of our approach, we analyzed the EEG data that was obtained from a modified Sternberg memory experiment. Two almost uncorrelated components obtained by applying the sparse factorization method were selected for phase synchronization analysis. Several interesting findings were obtained, especially that memory-related synchronization and desynchronization appear in the alpha band, and that the strength of alpha band synchronization is related to memory performance. Index Terms--Electroencephalogram (EEG), linear mixture, linear programming, sparse factorization, synchronization, wavelet packets.

In this letter, we solve the problem of identifying matrices S [member of] [R.sup.n x N] and A [member of] [R.sup.m x n] knowing only their multiplication X = AS, under some conditions, expressed either in terms of A and sparsity of S (identifiability conditions), or in terms of X (sparse component analysis (SCA) conditions). We present algorithms for such identification and illustrate them by examples. Index Terms--Blind source separation (BSS), sparse component analysis (SCA), underdetermined mixtures.

This paper reports a study on the problem of the blind simultaneous extraction of specific groups of independent components from a linear mixture. This paper first presents a general overview and unification of several information theoretic criteria for the extraction of a single independent component. Then, our contribution fills the theoretical gap that exists between extraction and separation by presenting tools that extend these criteria to allow the simultaneous blind extraction of subsets with an arbitrary number of independent components. In addition, we analyze a family of learning algorithms based on Stiefel manifolds and the natural gradient ascent, present the nonlinear optimal activations (score) functions, and provide new or extended local stability conditions. Finally, we illustrate the performance and features of the proposed approach by computer-simulation experiments. Index Terms--Blind-signal extraction, blind signal separation, independent component analysis, negentropy and minimum entropy, projection pursuit.

Independent component analysis is to extract independent signals from their linear mixtures without assuming prior knowledge of their mixing coefficients. As we know, a number of factors are likely to affect separation results in practical applications, such as the number of active sources, the distribution of source signals, and noise. The purpose of this paper to develop a general framework of blind separation from a practical point of view with special emphasis on the activation function adaptation. First, we propose the exponential generative model for probability density functions. A method of constructing an exponential generative model from the activation functions is discussed. Then, a learning algorithm is derived to update the parameters in the exponential generative model. The learning algorithm for the activation function adaptation is consistent with the one for training the demixing model. Stability analysis of the learning algorithm for the activation function is also discussed. Both theoretical analysis and simulations show that the proposed approach is universally convergent regardless of the distributions of sources. Finally, computer simulations are given to demonstrate the effectiveness and validity of the approach. Index Terms--Activation function, blind source separation, exponential family, independent component analysis.

In this paper, we propose a robust approach for independent component analysis (ICA) of signals that observations are contaminated with high-level additive noise and/or outliers. The source signals may contain mixtures of both sub-Gaussian and super-Gaussian components, and the number of sources is unknown. Our robust approach includes two procedures. In the first procedure, a robust prewhitening technique is used to reduce the power of additive noise, the dimensionality and the correlation among sources. A cross-validation technique is introduced to estimate the number of sources in this first procedure. In the second procedure, a nonlinear function is derived using the parameterized t-distribution density model. This nonlinear function is robust against the undue influence of outliers fundamentally. Moreover, the stability of the proposed algorithm and the robust property of misestimating the parameters (kurtosis) have been studied. By combining the t-distribution model with a family of light-tailed distributions (sub-Gaussian) model, we can separate the mixture of sub-Gaussian and super-Gaussian source components. Through the analysis of artificially synthesized data and real-world magnetoencephalographic (MEG) data, we illustrate the efficacy of this robust approach. Index Terms--Cross-validation method, independent component analysis (ICA), parametric estimation method, principal component analysis (PCA), robust prewhitening, t-distribution density model, unaveraged single-trial MEG data analysis.

This paper presents the derivation of an unsupervised learning algorithm, which enables the identification and visualization of latent structure within ensembles of high-dimensional data. This provides a linear projection of the data onto a lower dimensional subspace to identify the characteristic structure of the observations independent latent causes. The algorithm is shown to be a very promising tool for unsupervised exploratory data analysis and data visualization. Experimental results confirm the attractiveness of this technique for exploratory data analysis and an empirical comparison is made with the recently proposed generative topographic mapping (GTM) and standard principal component analysis (PCA). Based on standard probability density models a generic nonlinearity is developed which allows both 1) identification and visualization of dichotomised clusters inherent in the observed data and 2) separation of sources with arbitrary distributions from mixtures, whose dimensionality may be greater than that of number of sources. The resulting algorithm is therefore also a generalized neural approach to independent component analysis (ICA) and it is considered to be a promising method for analysis of real-world data that will consist of sub- and super-Gaussian components such as biomedical signals. Index Terms - Blind source separation, data clustering, data visualization, independent component analysis, unsupervised learning.

In this paper a new class of simplified low-cost analog artificial neural networks with on chip adaptive learning algorithms are proposed for solving linear systems of algebraic equations in real time. The proposed learning algorithms for linear least squares (LS), total least squares (TLS) and data least squares (DLS) problems can be considered as modifications and extensions of well known algorithms: the row-action projection-Kaczmarz algorithm [25] and/or the LMS (Adaline) Widrow-Hoff algorithms [21]. The algorithms can be applied to any problem which can be formulated as a linear regression problem. The correctness and high performance of the proposed neural networks are illustrated by extensive computer simulation results.

In this paper we present an iterative inversion (II) approach to blind source separation (BSS). It consists of a quasi-Newton method for the resolution of an estimating equation obtained from the implicit inversion of a robust estimate of the mixing system. The resulting learning rule includes several existing algorithms for BSS as particular cases giving them a novel and unified interpretation. It also provides a justification of the Cardoso and Laheld step size normalization [12]. The II method is first presented for instantaneous mixtures and then extended to the problem of blind separation of convolutive mixtures. Finally, we derive the necessary and sufficient asymptotic stability conditions for both the instantaneous and convolutive methods to converge. Index Terms--Adaptive signal processing, blind source separation (BSS), equivariant learning algorithms, higher order statistics, independent component analysis (ICA), multichannel blind deconvolution (MBD).

Nonnegative matrix factorization (NMF) is an unsupervised learning method useful in various applications including image processing and semantic analysis of documents. This paper focuses on symmetric NMF (SNMF), which is a special case of NMF decomposition. Three parallel multiplicative update algorithms using level 3 basic linear algebra subprograms directly are developed for this problem. First, by minimizing the Euclidean distance, a multiplicative update algorithm is proposed, and its convergence under mild conditions is proved. Based on it, we further propose another two fast parallel methods: \alpha-SNMF and \beta-SNMF algorithms. All of them are easy to implement. These algorithms are applied to probabilistic clustering. We demonstrate their effectiveness for facial image clustering, document categorization, and pattern clustering in gene expression. [ABSTRACT FROM PUBLISHER]

We propose a novel efficient method of blind signal extraction from multi-sensor networks when each observed signal consists of one global signal and local uncorrelated signals. Most of existing blind signal separation and extraction methods such as independent component analysis have constraints such as statistical independence, non-Gaussianity, and underdetermination, and they are not suitable for global signal extraction problem from noisy observations. We developed an estimation algorithm based on alternating iteration and the smart weighted averaging. The proposed method does not have strong assumptions such as independence or non-Gaussianity. Experimental results using a musical signal and a real electroencephalogram demonstrate the advantage of the proposed method. [ABSTRACT FROM AUTHOR]

This paper discusses the estimation and numerical calculation of the probability that the 0-norm and 1-norm solutions of underdetermined linear equations are equivalent in the case of sparse representation. First, we define the sparsity degree of a signal. Two equivalence probability estimates are obtained when the entries of the 0-norm solution have different sparsity degrees. One is for the case in which the basis matrix is given or estimated, and the other is for the case in which the basis matrix is random. However, the computational burden to calculate these probabilities increases exponentially as the number of columns of the basis matrix increases. This computational complexity problem can be avoided through a sampling method. Next, we analyze the sparsity degree of mixtures and establish the relationship between the equivalence probability and the sparsity degree of the mixtures. This relationship can be used to analyze the performance of blind source separation (BSS). Furthermore, we extend the equivalence probability estimates to the small noise case. Finally, we illustrate how to use these theoretical results to guarantee a satisfactory performance in underdetermined BSS. [ABSTRACT FROM AUTHOR]

In this paper, we use a two-stage sparse factorization approach for blindly estimating the channel parameters and then estimating source components for electroencephalogram (EEG) signals. EEG signals are assumed to be linear mixtures of source components, artifacts, etc. Therefore, a raw EEG data matrix can be factored into the product of two matrices, one of which represents the mixing matrix and the other the source component matrix. Furthermore, the components are sparse in the time-frequency domain, i.e., the factorization is a sparse factorization in the time frequency domain. It is a challenging task to estimate the mixing matrix. Our extensive analysis and computational results, which were based on many sets of EEG data, not only provide firm evidences supporting the above assumption, but also prompt us to propose a new algorithm for estimating the mixing matrix. After the mixing matrix is estimated, the source components are estimated in the time frequency domain using a linear programming method. In an example of the potential applications of our approach, we analyzed the EEG data that was obtained from a modified Sternberg memory experiment. Two almost uncorrelated components obtained by applying the sparse factorization method were selected for phase synchronization analysis. Several interesting findings were obtained, especially that memory-related synchronization and desynchronization appear in the alpha band, and that the strength of alpha band synchronization is related to memory performance. [ABSTRACT FROM AUTHOR]

Independent component analysis is to extract independent signals from their linear mixtures without assuming prior knowledge of their mixing coefficients. As we know, a number of factors are likely to affect separation results in practical applications, such as the number of active sources, the distribution of source signals, and noise. The purpose of this paper to develop a general framework of blind separation from a practical point of view with special emphasis on the activation function adaptation. First, we propose the exponential generative model for probability density functions. A method of constructing an exponential generative model from the activation functions is discussed. Then, a Learning algorithm is derived to update the parameters in the exponential generative model. The learning algorithm for the activation function adaptation is consistent with the one for training the denting model. Stability analysis of the learning algorithm for the activation function is also discussed. Both theoretical analysis and simulations show that the proposed approach is universally convergent regardless of the distributions of sources. Finally, computer simulations are given to demonstrate the effectiveness and validity of the approach. [ABSTRACT FROM AUTHOR]

This article introduces the present issue of the journal "IEEE Transactions on Neural Networks." Nonlinear signal processing using neural networks inherited the dominant mean-squared error criteria from linear adaptive filtering theory due to a number of appealing properties exhibited by second-order statistical optimality criteria under the linear model. Information theory has a long history in the design of optimal communication systems and coding. Numerous useful theoretical results relating information theoretic learning (ITL). The aim of this special issue is to present the current state of the art in the application of information theoretic concepts and approaches to learning, adaptation, and neural network theories by active experts working in the broadly defined area of ITL.