Independent Component Analysis.

Independent component analysis (ICA) aims at decomposing an observed random vector into statistically independent variables Deflation-based implementations, such as the popular one-unit FastICA algorithm and its variants, extract the independent components one after another A novel method for deflationary ICA, referred to as RobustICA, is put forward in this paper This simple technique consists of performing exact line search optimization of the kurtosis contrast function The step size leading to the global maximum of the contrast along the search direction is found among the roots of a fourth-degree polynomial This polynomial rooting can be performed algebraically, and thus at low cost, at each iteration Among other practical benefits, RobustICA can avoid prewhitening and deals with real- and complex-valued mixtures of possibly noncircular sources alike The absence of prewhitening improves asymptotic performance The algorithm is robust to local extrema and shows a very high convergence speed in terms of the computational cost required to reach a given source extraction quality, particularly for short data records These features are demonstrated by a comparative numerical analysis on synthetic data RobustICA's capabilities in processing real-world data involving noncircular complex strongly super-Gaussian sources are illustrated by the biomedical problem of atrial activity (AA) extraction in atrial fibrillation (AF) electrocardiograms (ECGs), where it outperforms an alternative ICA-based technique

/pdf/robust-independent-component-analysis-by-iterative-42vorvxx4k.pdf

Robust Independent Component Analysis by Iterative Maximization of the Kurtosis Contrast With Algebraic Optimal Step Size

In this paper, the conditions for identifiability, separability and uniqueness of linear complex valued independent component analysis (ICA) models are established. These results extend the well-known conditions for solving real-valued ICA problems to complex-valued models. Relevant properties of complex random vectors are described in order to extend the Darmois-Skitovich theorem for complex-valued models. This theorem is used to construct a proof of a theorem for each of the above ICA model concepts. Both circular and noncircular complex random vectors are covered. Examples clarifying the above concepts are presented

Complex random vectors and ICA models: identifiability, uniqueness, and separability

In this paper, we present a partial survey of the tools borrowed from tensor algebra, which have been utilized recently in Statistics and Signal Processing. It is shown why the decompositions well known in linear algebra can hardly be extended to tensors. The concept of rank is itself difficult to define, and its calculation raises difficulties. Numerical algorithms have nevertheless been developed, and some are reported here, but their limitations are emphasized. These reports hopefully open research perspectives for enterprising readers.

https://hal.archives-ouvertes.fr/hal-00347139/document

Tensor Decompositions, State of the Art and Applications

A communications device for separating source signals provided by M signal sources includes an antenna array comprising N antenna elements for receiving at least N different summations of the M source signals. A respective in-phase and quadrature module is connected downstream to each antenna element for separating each one of the N different summations of the M source signals received thereby into an in-phase and quadrature component set. A blind signal separation processor forms a mixing matrix comprising at least 2N different summations of the M source signals, with each in-phase and quadrature component set providing 2 inputs into the mixing matrix. The mixing matrix has a rank equal up to 2N. The desired source signals are separated from the mixing matrix by the blind signal separation processor.

Blind signal separation using I and Q components

A polynomial criterion is proposed to perform blind source separation, extending previous works to MIMO systems. The criterion is proved to be asymptotically MAP-equivalent in presence of PSK sources. An efficient minimization algorithm dedicated to polynomial criteria is then developed, improving on the fixed-step stochastic gradient previously utilized in this framework.

/pdf/blind-separation-of-discrete-sources-1nxq5y2fjs.pdf

Blind separation of discrete sources

A novel analytical blind single-input single-output (SISO) identification algorithm is presented, based on the noncircular second-order statistics of the output. It is shown that statistics of order higher than two are not mandatory to restore identifiability. Our approach is valid, for instance, when the channel is excited by phase shift keying (PSK) inputs. It is shown that the channel taps need to satisfy a polynomial system of degree 2 and that identification amounts to solving the system. We describe the algorithm that is able to solve this particular system entirely analytically, thus avoiding local minima. Computer results eventually show the robustness with respect to noise and to channel length overdetermination. Identifiability issues are also addressed.

/pdf/analytical-blind-channel-identification-369brxp6kn.pdf

Analytical blind channel identification

In this paper, a new blind source separation algorithm is described. Its main feature is to be analytical, in other words it doesn't suffer from local minima. The proposed method uses the discrete character of digital sources, which yields a polynomial system. The estimation of the sources is shown to be equivalent to the computation of rank-one tensors that are found by means of the old method by Macaulay for the computation of resultant. Finally, computer simulations are presented and the performances are compared to the analytical CM algorithm by van der Veen.

Analytical blind discrete source separation

Many blind or semi-blind equalizers are implemented with the help of iterative algorithms, and therefore may require long convergence times or suffer from local minima. Closed-form block blind equalizers are attractive, even if they are suboptimal, since they can serve to initialize them. But they can perform better in time-varying contexts, that is on very short data blocks. The authors focused mainly on PSK modulations, including MSK and QPSK. The proposed algorithms are based on known instantaneous properties of such modulations.

Closed-form equalization

Blind equalization of non-minimum phase FIR channels requires prior identification, for stability reasons. We present a novel algorithm that is able to identify a channel in the presence of an unknown MSK modulated input (which can be viewed as an approximation of the GMSK modulation used in GSM mobile systems), by resorting only to output second order moments. Blind identification is made possible because the input is not circular. It is shown that this approach leads to a system of L quadrics in L unknowns, if L denotes the number of taps of the unknown FIR channel. This system is then solved with the help of an original algorithm based on the resultant techniques. The performances in terms of bit error rates are also reported.

O. Grellier

Papers

Blind separation of discrete sources

Analytical blind channel identification

Analytical blind discrete source separation

Closed-form equalization

Closed-form blind channel identification with MSK inputs