Catálogo de publicaciones - libros

The discussion in this paper revolves around the notion of . The latter can be thought of as a unifying concept which includes a variety of important problems in applied mathematics. Thus, for example, the problems of classification, clustering, image segmentation, and discriminant analysis can all be regarded as separation problems in which one is looking for a decision boundary to be used in order to separate a set of data points into a number of (homogeneous) subsets described by different conditional densities. Since, in this case, the decision boundary can be defined as a hyperplane, the related separation problems can be regarded as . On the other hand, the problems of source separation, deconvolution, and independent component analysis represent another subgroup of separation problems which address the task of separating mixed signals. The main idea behind the present development is to show conceptually and experimentally that both geometric and algebraic separation problems are very intimately related, since there exists a general variational approach based on which one can recover either geometrically or algebraically mixed sources, while only little needs to be modified to go from one setting to another.

Delayed mixing is a problem of theoretical interest and practical importance, e.g., in speech processing, bio-medical signal analysis and financial data modelling. Most previous analyses have been based on models with integer shifts, i.e., shifts by a number of samples, and have often been carried out using time-domain representation. Here, we explore the fact that a shift in the time domain corresponds to a multiplication of in the frequency domain. Using this property an algorithm in the case of sources≤sensors allowing arbitrary mixing and delays is developed. The algorithm is based on the following steps: 1) Find a subspace of shifted sources. 2) Resolve shift and rotation ambiguity by information maximization in the complex domain. The algorithm is proven to correctly identify the components of synthetic data. However, the problem is prone to local minima and difficulties arise especially in the presence of large delays and high frequency sources. A Matlab implementation can be downloaded from [1].

We extend the Gaussian scale mixture model of dependent subspace source densities to include non-radially symmetric densities using Generalized Gaussian random variables linked by a common variance. We also introduce the modeling of skew in source densities and subspaces using a generalization of the Normal Variance-Mean mixture model. We give closed form expressions for subspace likelihoods and parameter updates in the EM algorithm.

In recent years, there has been an increasing interest in developing new algorithms for digital signal processing by applying and generalising existing numerical linear algebra tools. A recent result shows that the FastICA algorithm, a popular state-of-the-art method for linear Independent Component Analysis (ICA), shares a nice interpretation as a Newton type method with the Rayleigh Quotient Iteration (RQI), the latter method wellknown to the numerical linear algebra community. In this work, we develop an analogous theory of single vector iteration ICA methods. Two classes of methods are proposed for the one-unit linear ICA problem, namely, power ICA methods and inverse iteration ICA methods. By means of a , scalar shifted versions of both power ICA method and inverse iteration ICA method are proposed and proven to be locally quadratically convergent to a correct demixing vector.

The applications of Non-Negative Tensor Factorization (NNTF) is an important tool for brain wave (EEG) analysis. For it to work efficiently, it is essential for NNTF to have a unique solution. In this paper we give a sufficient condition for NNTF to have a unique global optimal solution. For a third-order tensor we define a matrix by some rearrangement of and it is shown that the rank of the matrix is less than or equal to the rank of . It is also shown that if both ranks are equal to , the decomposition into a sum of tensors of rank 1 is unique under some assumption.

With the advent of high-throughput data recording methods in biology and medicine, the efficient identification of meaningful subspaces within these data sets becomes an increasingly important challenge. Classical dimension reduction techniques such as principal component analysis often do not take the large statistics of the data set into account, and thereby fail if the signal space is for example of low power but meaningful in terms of some other statistics. With ‘colored subspace analysis’, we propose a method for linear dimension reduction that evaluates the time structure of the multivariate observations. We differentiate the signal subspace from noise by searching for a subspace of non-trivially autocorrelated data; algorithmically we perform this search by joint low-rank approximation. In contrast to blind source separation approaches we however do not require the existence of sources, so the model is applicable to any wide-sense stationary time series without restrictions. Moreover, since the method is based on second-order time structure, it can be efficiently implemented even for large dimensions. We conclude with an application to dimension reduction of functional MRI recordings.

Renyi’s entropy-based criterion has been proposed as an objective function for independent component analysis because of its relationship with Shannon’s entropy and its computational advantages in specific cases. These criteria were suggested based on “convincing” experiments. However, there is no theoretical proof that globally maximizing those functions would lead to separate the sources; actually, this was implicitly conjectured. In this paper, the problem is tackled in a theoretical way; it is shown that globally maximizing the Renyi’s entropy-based criterion, in its general form, does not necessarily provide the expected independent signals. The contrast function property of the corresponding criteria simultaneously depend on the value of the Renyi parameter, and on the (unknown) source densities.

In this paper we propose a Variational Bayesian (VB) estimation approach for Blind Sources Separation (BSS) problem, as an alternative method to MCMC. The data are images and the sources are images which are assumed piecewise homogeneous. To insure these properties, we propose a piecewise Gauss-Markov model for the sources with a hidden classification variable which is modeled by a Potts-Markov field. A few simulation results are given to illustrate the performances of the proposed method and some comparison with other methods (MCMC and VBICA) used for BSS, are presented.

Cyclostationary signals can be met in various domains, such as telecomunications and vibration analysis. Cyclostationarity allows to model repetitive signals and hidden periodicities such as those induced by modulation for communications and by rotating machines for vibrations. In some cases, the fundamental frequency of these repetitive phenomena can be known. The algorithm that we propose aims at extracting one cyclostationary source, whose cyclic frequency is known, from a set of observations. We propose a new criterion based on second order statistics of the measures which is easy to estimate and leads to extraction with very good accuracy.

In this paper, we propose to use the Huber -estimator cost function as a contrast function within the complex FastICA algorithm of Bingham and Hyvarinen for the blind separation of mixtures of independent, non-Gaussian, and proper complex-valued signals. Sufficient and necessary conditions for the local stability of the complex-circular FastICA algorithm for an arbitrary cost are provided. A local stability analysis shows that the algorithm based on the Huber -estimator cost has behavior that is largely independent of the cost function’s threshold parameter for mixtures of non-Gaussian signals. Simulations demonstrate the ability of the proposed algorithm to separate mixtures of various complex-valued sources with performance that meets or exceeds that obtained by the FastICA algorithm using kurtosis-based and other contrast functions.