# HANDBOOK OF BLIND SOURCE SEPARATION PDF

Comon, Pierre. Handbook of blind source separation: independent component analysis and blind deconvolution. 1. probability density function (pdf), download Handbook of Blind Source Separation - 1st Edition. Print Book & E- Book. DRM-free (EPub, PDF, Mobi). × DRM-Free Easy - Download and start. Handbook of Blind Source Separation,. Independent Component Analysis and Applications. P. Comon and C. Jutten Eds. October 8,

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Handbook of Blind Source Separation. Independent ICA techniques in the signal processing community and identify those that are. Handbook of Blind Source Separation: Independent Component Analysis and Applications [Pierre Comon, Christian Jutten] on computerescue.info *FREE* shipping . Handbook of Blind Source Separation,. Independent Component Analysis and Applications. P. Comon and C. Jutten Eds. October 6,

Karhunen Semi-blind methods for communications -- V.

Zarzoso, P. Comon, and D.

## Handbook of Blind Source Separation

Deville, C. Jutten, and R. Chevalier and A. Albera, P. Comon, L. Parra, A.

Karfoul, A. Kachenoura, and L. Vincent and Y.

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BTDs write a given tensor as a sum of terms that have low multilinear rank, without having to be rank In this paper we explain how BTDs can be used for factor analysis and blind source separation.

Different variants of the approach are illustrated with examples. Preview Unable to display preview. Download preview PDF. References 1. Bruckstein, A. SIAM Rev. Candes, E. IEEE Trans. Carroll, J.

When context 2 blue in Fig. An important point was revealed at the first step of the third session, in which context 1 was provided again. The BSS error was significantly smaller than that in the first session and close to zero from the beginning of this session, indicating that the network retained synaptic strengths that were optimized for context 1 even after learning context 2.

After several iterations, the BSS error for both contexts converged to zero. The success of learning was also confirmed by the trajectory of the EGHR cost function that also converged to the minimum value Fig. These results show that an undercomplete EGHR increased the speed of re-adaptation to previously experienced contexts, suggesting that memory of past experiences was preserved within the network.

Moreover, the network learned the optimal set of synaptic strengths that entertained both contexts after several iterations.

This freedom spanned a null space in which synaptic strengths were equally optimized with zero BSS error. Similarly, when two different contexts were considered, four-dimensional degrees of freedom remained, as an overlap between the two eight-dimensional null spaces.

To visualize such a null space, we projected synaptic strengths onto a subspace spanned by the first PC1 and second PC2 principal components of the trajectory of synaptic strengths Fig. On this PC1-PC2 plane, a null space was illustrated as a nullcline.

Since the dynamics of synaptic strengths were determined to go down the slope of a cost function for either context 1 or 2, synaptic strengths were started from a random initial state and reached the nullcline of either context 1 or 2, in turn.

Because of this, the BSS error reached zero after iterative training; i. Our agent received redundant dimensional sensory inputs, comprising sets contexts of mixtures of ten hidden sources sources in total , that were generated as products of the context-dependent mixing matrix and sources.

Ten outputs neurons learned to infer each source from their mixtures by updating synaptic strengths through the EGHR. After training, we found that they successfully represented the ten sources for every context, without further updating synaptic strengths, as illustrated by the reduction of the BSS error for all contexts Fig.

This was because synaptic strengths had sufficient capacity and were formed to express the inverse of the concatenated mixing matrices from all contexts, which was further confirmed by the convergence of the synaptic strength matrix in the null space Fig. Figure 3 BSS with large number of contexts.

One of different contexts was randomly selected for each session. In each session, the dimensional sensory inputs x were generated from ten-dimensional hidden sources s , which independently followed the unit Laplace distribution, through a context-dependent random mixing matrix A k. The shaded area shows the standard deviation. B Mappings from ten sources to ten outputs in contexts 1 and after training.

## Multi-context blind source separation by error-gated Hebbian rule

C The dynamics of synaptic strength matrix W projected in the three-dimensional space spanned by the first to third principal components PC1 to PC3. The matrix starts from a random initial point star mark and converges to the null space, in which synaptic strengths are optimized for all trained contexts. Full size image BSS in constantly time-varying environments In the previous section, we described a general condition for the neural network to achieve the multi-context BSS.

Here, we show that when contexts are generated from a low-dimensional subspace of mixing matrices and, therefore, are dependent on each other, the EGHR can find the common features and use them to perform the multi-context BSS.

Each component of R t is supposed to on average slowly change, i. This condition is required to distinguish whether changes in inputs are caused by changes in the mixing matrix A t or the hidden sources s t. Formally, A t expresses infinite contexts along the trajectory of R t.

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This is a more complicated setup than the standard BSS in the sense that both sources and the mixing matrix change in time. The above condition means that the network performs BSS based on the time-invariant features A 0 of the mixing matrix, while neglecting the time-varying features A 1 R t.

This can be viewed as a way to compress high-dimensional data. This is distinct from the standard dimensionality reduction approach by PCA, which would preferentially extract the time-variant features due to their extra variances. Moreover, the ability to perform dimensionality reduction is an important advantage of the EGHR over conventional ICA algorithms, such as the infomax-based ICA 10 , 11 , natural gradient 12 and nonholonomic 39 algorithms, and the ICA mixture model 40 , because these learning algorithms do not learn effective dimensionality reduction in the multi-context BSS setup due to their construction see Methods for mathematical explanations.

Figure 4 BSS with time-varying mixing matrix. A Top: Schematic image of sensory inputs generated from two sources through time-varying mixing matrix A t. The mixing matrix is controlled by the low-dimensional rotation matrix R t. Bottom: Trajectories of hidden sources and an element of R t , showing the difference in their time courses. B Trajectory of BSS error. C Trajectories of mapping weights from sources to outputs, i.

The matrix starts from a random initial point star mark and follows a spiral trajectory as it converges to a subspace in which synaptic matrix W is perpendicular to the time-varying component A 1. E Overlap of synaptic matrix W with time-invariant component A 0 and time-variant component A 1.

## Book chapter

The overlap between two matrices was defined by the Frobenius norm of their product, i. The simulation showed a reduction in the BSS error Fig. As illustrated in Fig. After training, the overlap converged to zero. Hence, synaptic strengths were optimized regardless of R t at this solution, which enabled the network to perform BSS with a virtually infinite number of contexts. Indeed, a mathematical analysis shows that multi-context BSS is possible for a general time-varying matrix R t as long as it changes slowly enough see Methods.

Next, we demonstrated the utility of the EGHR, when supplied with redundant inputs, by using natural birdsongs and a time-variant mixing matrix that expressed a natural contextual change.

To obtain time-independent features, we assumed that the two birds moved around in non-overlapping areas. For simplicity, we also assumed that the two birds moved around at different heights. The agent received mixtures of the two birdsongs through six microphones with different direction preferences. In the current context, the z-axis of the birds was time-invariant and the x- and y-axes of the birds were time-variant, although the observer was not informed about this.

By tuning synaptic strengths by the EGHR, neural outputs were established to infer each birdsong, while the mixing matrix changed continuously.

Hence, neural outputs could separate the two birdsongs, although the amplitudes of the songs recorded by the microphones continuously changed depending on the positions of birds. Figure 5 BSS of birdsongs when two birds move around the agent.Overlap between two matrices is defined by WA k F, as described in Fig.

Institutional Subscription. Directions of work or proceedings. It is also crucial for animals to generalize past learning to inexperienced contexts. Similarly, when two different contexts were considered, four-dimensional degrees of freedom remained, as an overlap between the two eight-dimensional null spaces.

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