Schur\u27s Lemma For Coupled Reducibility And Coupled Normality

Let $\mathcal A = \{A_{ij} \}_{i, j \in \mathcal I}$, where $\mathcal I$ is an index set, be a doubly indexed family of matrices, where $A_{ij}$ is $n_i \times n_j$. For each $i \in \mathcal I$, let $\mathcal V_i$ be an $n_i$-dimensional vector space. We say $\mathcal A$ is reducible in the coupled sense if there exist subspaces, $\mathcal U_i \subseteq \mathcal V_i$, with $\mathcal U_i \neq \{0\}$ for at least one $i \in \mathcal I$, and $\mathcal U_i \neq \mathcal V_i$ for at least one $i$, such that $A_{ij} (\mathcal U_j) \subseteq \mathcal U_i$ for all $i, j$. Let $\mathcal B = \{B_{ij} \}_{i, j \in \mathcal I}$ also be a doubly indexed family of matrices, where $B_{ij}$ is $m_i \times m_j$. For each $i \in \mathcal I$, let $X_i$ be a matrix of size $n_i \times m_i$. Suppose $A_{ij} X_j = X_i B_{ij}$ for all $i, j$. We prove versions of Schur\u27s lemma for $\mathcal A, \mathcal B$ satisfying coupled irreducibility conditions. We also consider a refinement of Schur\u27s lemma for sets of normal matrices and prove corresponding versions for $\mathcal A, \mathcal B$ satisfying coupled normality and coupled irreducibility conditions

A neuromimetic solution for the problem of sources discrimination

The problem of sources discrimination, very classical in Signal Processing field, is also an actual problem in biological systems . Biological sensors are sensitive to many sources, so the Central Nervous System processes typically multidimensional signais, each component of which is an unknown mixture of unknown sources, assumed independent . The neuromimetic solution, proposed in this paper, is based on a recursive and fully-interconnected network of operators. The weight of the connections are varying according to an adaptation rule, which executes an independence test of the network outputs . With regard to adaptation rudes used in adaptive filtering, here the adaptive increment is achieved necessarily by the product of two non-linear functions . Some experimental results, in Signal Processing and Image Processing fields, show the efficiency of this adaptive algorithm . We proue also the possible generalization of this algorithm in the case of more complex (non-linear, degenerated, etc .) mixtures . This algorithm points out a new concept of Independent Components Analysis, more powerful than this one of Principal Components Analysis, and applicable in the general frame of data analysis.Le problème de séparation de sources, très classique en traitement du signal, correspond aussi à une réalité dans les systèmes biologiques . En effet, les capteurs biologiques sont sensibles à de multiples sources, le système nerveux central traite donc des signaux multi-capteurs dont chaque composante est un mélange inconnu de sources inconnues, supposées indépendantes . La solution neuromimétique, proposée dans cet article est fondée sur un réseau récursif d'opéreurs complètement interconnectés, et dont le poids des connexions évoluent selon une règle d'adaptation, qui opère un test d'indépendance des sorties du réseau . Par rapport aux règles utilisées en filtrage adaptatif, l'incrément d'adaptation fait intervenir nécessairement le produit de deux fonctions non linéaires. Plusieurs résultats expérimentaux dans le domaine du Traitement du Signal ou du Traitement d'Images illustrent les performances de l'algorithme . La généralisation à des mélanges plus complexes, ou dégénérés est également discutée et illustrée . Cet algorithme révèle aussi un nouveau concept d'Analyse en Composantes Indépendantes, plus fort que celui d'Analyse en Composantes Principales, applicable dans le cadre général de l'analyse de données

Using state space differential geometry for nonlinear blind source separation

Given a time series of multicomponent measurements of an evolving stimulus, nonlinear blind source separation (BSS) seeks to find a "source" time series, comprised of statistically independent combinations of the measured components. In this paper, we seek a source time series with local velocity cross correlations that vanish everywhere in stimulus state space. However, in an earlier paper the local velocity correlation matrix was shown to constitute a metric on state space. Therefore, nonlinear BSS maps onto a problem of differential geometry: given the metric observed in the measurement coordinate system, find another coordinate system in which the metric is diagonal everywhere. We show how to determine if the observed data are separable in this way, and, if they are, we show how to construct the required transformation to the source coordinate system, which is essentially unique except for an unknown rotation that can be found by applying the methods of linear BSS. Thus, the proposed technique solves nonlinear BSS in many situations or, at least, reduces it to linear BSS, without the use of probabilistic, parametric, or iterative procedures. This paper also describes a generalization of this methodology that performs nonlinear independent subspace separation. In every case, the resulting decomposition of the observed data is an intrinsic property of the stimulus' evolution in the sense that it does not depend on the way the observer chooses to view it (e.g., the choice of the observing machine's sensors). In other words, the decomposition is a property of the evolution of the "real" stimulus that is "out there" broadcasting energy to the observer. The technique is illustrated with analytic and numerical examples.Comment: Contains 14 pages and 3 figures. For related papers, see http://www.geocities.com/dlevin2001/ . New version is identical to original version except for URL in the bylin

Robustesse des hypothèses dans une méthode sous-espace pour la séparation de sources.

Les méthodes de type sous-espace pour la séparation de sources supposent théoriquement la connaissance exacte de l'ordre des filtres du mélange. Dans ce papier, on montre expérimentalement que la séparation est possible si l'ordre est sous-estimé, pourvu que cet ordre soit suffisant pour modéliser fonctionnellement les filtres du mélange

Use of Bimodal Coherence to Resolve Spectral Indeterminacy in Convolutive BSS

Recent studies show that visual information contained in visual speech can be helpful for the performance enhancement of audio-only blind source separation (BSS) algorithms. Such information is exploited through the statistical characterisation of the coherence between the audio and visual speech using, e.g. a Gaussian mixture model (GMM). In this paper, we present two new contributions. An adapted expectation maximization (AEM) algorithm is proposed in the training process to model the audio-visual coherence upon the extracted features. The coherence is exploited to solve the permutation problem in the frequency domain using a new sorting scheme. We test our algorithm on the XM2VTS multimodal database. The experimental results show that our proposed algorithm outperforms traditional audio-only BSS

Quelques remarques méthodologiques concernant la séparation des mélanges convolutifs par l'approche de décorrélation

Dans cet article, nous étudions un problème de séparation de mélange convolutif 2 sources / 2 capteurs. Nous étudions la pertinence de la technique dite de décorrélation, consistant à séparer le mélange en adaptant un filtre qui, appliqué aux observations, génère des signaux décorrélés entre eux. Dans le cas d'un filtre de mélange RIF, cette technique permet en théorie de séparer les sources. Notre contribution établit que si le filtre de mélange est RII, les équations exprimant la décorrélation admettent une infinité de solutions dont nous donnons une paramétrisation. Afin de remédier à cet inconvénient, nous suggérons l'utilisation d'une technique basée sur la prédiction linéaire

Fast Approximation of Nonlinearities for improving inversion algorithms of PNL mixtures and Wiener systems

This paper proposes a very fast method for blindly approximating a nonlinear mapping which transforms a sum of random variables. The estimation is surprisingly good even when the basic assumption is not satisfied.We use the method for providing a good initialization for inverting post-nonlinear mixtures and Wiener systems. Experiments show that the algorithm speed is strongly improved and the asymptotic performance is preserved with a very low extra computational cost