Analyse en Composantes Indépendantes Multidimensionnelles via des cumulants d’ordres variés

The author deals with the problem of multidimensional independent component analysis (MICA) which is the natural generalization of the ordinary problem of independent component analysis (ICA). First, in order to facilitate the use of higher-order cumulants, we present new formulas for the cumulant matrices of a real random vector from its moment matrices. In addition to the usual matrix operations, these formulas use only the Kronecker product, the vec operator and some commutation matrices. These formulas lend themselves to examine more closely the specific structures of cumulant matrices and provide results on the ranks of these matrices that characterize the dependence between random variables composing the random vector. The main practical interest of our matrix formulas lies in much easier cumulant evaluation and faster computation than the conventional method based on repeated use of the Leonov and Shiryaev formulas. In the second part of this thesis, we show that under the usual assumptions of the independent multidimensional component analysis, contracted cumulant matrices at any statistical order are all block diagonalizable in the same basis. We derive an algorithm for solving MICA by block diagonalization and compare the results obtained to the orders 3-6, between them and with other methods, on several synthetic signals. Simple examples are developed to justify the need to combine different levels to ensure the best separation. We also prove that the easiest case to deal with is the case of mixtures of sources that have different dimensions. In the last part of this thesis we propose a set of methods that operate only the higher- order statistics. Under certain additional assumptions, these methods are shown to completely solve the noisy MICA problem without second-order whitening by joint block diagonalization of a cumulant matrices set coming from statistics of orders strictly higher than four. A comparison with the second-order based whitening MICA methods for the separation of fetal and maternal electrical activities (measured using three electrodes placed on the mother’s abdomen) shows that this new family is better suited to this application : it allow an almost perfect separation of these two contributions.L’auteur s’intéresse au problème de l’analyse en composantes indépendantes multidimensionnelles (ACIM) qui est la généralisation naturelle du problème ordinaire de l’analyse en composantes indépendantes (ACI). Tout d’abord, afin de faciliter l’utilisation des cumulants des ordres supérieurs, nous présentons de nou- velles formules pour le calcul matriciel des matrices de cumulants d’un vecteur aléatoire réel à partir de ses matrices de moments. Outre les opérations matricielles usuelles, ces formules utilisent uniquement le produit de Kronecker, l’opérateur vec et des matrices de commutation. Nous pouvons immédiatement à partir de ces formules examiner de plus près les structures particulières des matrices de cumulants et ainsi donner des résultats sur les rangs de ces matrices qui caractérisent la dépendance entre les variables aléatoires constituant le vecteur aléatoire. L’intérêt pratique principal de nos formules matricielles réside certainement dans une évaluation des cumulants beaucoup plus aisée et rapide qu’avec la méthode usuelle basée sur une utilisation répétée des formules de Leonov et Shiryaev. Dans la deuxième partie de cette thèse, nous montrons que sous les hypothèses usuelles de l’analyse en composantes indépendantes mul- tidimensionnelles, les matrices de cumulants contractées à un ordre statistique quelconque sont toutes bloc-diagonalisables dans la même base. Nous en déduisons des algorithmes de résolution d’ACIM par bloc-diagonalisation conjointe et comparons les résultats obtenus aux ordres 3 à 6, entre eux et avec d’autres méthodes, sur quelques signaux synthétiques. Des exemples simples ont élaborés afin de justifier la nécessité de combiner des ordres différents pour garantir la meilleure séparation. Nous prouvons aussi que le cas le plus simple à traiter est celui de mélanges de sources qui ont différentes dimensions. Dans la dernière partie de cette thèse nous proposons une famille de méthodes qui exploitent uniquement les sta- tistiques d’ordres supérieurs à deux. Sous certaines hypothèses supplémentaires, ces méthodes permettent après un blanchiment d’ordre quatre des observations de résoudre complètement le problème ACIM bruité en bloc diagonalisant conjointement un ensemble de matrices de cumulants issues des statistiques d’ordres supérieurs strictement à quatre. Une comparaison avec les méthodes ACIM à blanchiment d’ordre deux pour la séparation des activités électriques foetale et maternelle (mesurées via trois électrodes placées sur l’abdomen de la mère) montre que cette nouvelle famille est mieux adaptée à cette application : elles permettent une séparation quasi parfaite de ces deux contributions

Colored subspace analysis: Dimension reduction based on a signal's autocorrelation structure.

Identifying relevant signals within high-dimensional observations is an important preprocessing step for efficient data analysis. However, many classical dimension reduction techniques such as principal component analysis do not take the often rich statistics of real-world data into account, and thereby fail if for example the signal space is 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. We prove that the resulting signal subspace is uniquely determined by the data, given that all white components have been removed. Algorithmically we propose three efficient methods to perform this search, based on joint diagonalization, using a component clustering scheme, and via joint low-rank approximation. In contrast to temporal mixture approaches from blind signal processing we do not need a generative model, i.e., we 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 and applied even in large dimensions. Numerical examples together with an application to dimension reduction of functional MRI recordings demonstrate the usefulness of the proposed method. The implementation is publicly available as a Matlab package at http://cmb.helmholtz-muenchen.de/CSA

IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS: PART-I - Special Section on Blind Signal Processing and Its Applications

Blind signal processing (BSP) is currently one of the most exciting areas of research in statistical signal processing, unsupervised machine learning, neural networks, information theory, and exploratory data analysis. It has applications at the intersection of many science and engineering disciplines concerned with understanding and extracting useful information from data as diverse as neuronal activity and brain images, bioinformatics, communications, the World Wide Web, audio, video, and sensor signals. Because BSP is an interdisciplinary research area, the combination of ideas from the above disciplines is a developing avenue of research. The aim of this Special Section is to offer an opportunity to link these techniques in different areas and to find effectiveways of solving this problem. The Special Section constitutes a vehicle whereby researchers can present new studies of BSP, thus paving the way for future developments in the field.We received 20 submissions for consideration. After the review process, we selected the following eight papers for publication that span the approaches identified above. These are complex blind source extraction from noisy mixtures using second order statistics by Javidi et al.; complex independent component analysis by entropy bound minimization by Li et al.; real-time independent vector analysis for convolutive blind source separation by Kim; a nonnegative blind source separation model for binary test data by Schachtner et al.; a matrix pseudoinversion lemma and its application to block-based adaptive blind deconvolution for MIMO systems by Kohno et al.; colored subspace analysis: dimension reduction based on a signal’s autocorrelation structure by Theis; blind adaptive equalization of MIMO systems: new recursive algorithms and convergence analysis by Radenkovic et al.; and noise estimation using mean square cross prediction error for speech enhancement by Wang et al. We hope that this Special Section will stimulate interest in the challenging area of BSP, and look forward to seeing an increasing body of high-quality research aligned to this idea. We would like to express our gratitude to the authors of the papers in this special section, and also to the more than 60 reviewers who helped us evaluate the submissions