There is, I think, something ethereal about i —the square root of minus one. I remember first hearing about it at school. It seemed an odd beast at that time—an intruder hovering on the edge of reality.

Usually familiarity dulls this sense of the bizarre, but in the case of i it was the reverse: over the years the sense of its surreal nature intensified. It seemed that it was impossible to write mathematics that described the real world in …

I and i

https://www2.econ.iastate.edu/tesfatsi/DeepLearningInNeuralNetworksOverview.JSchmidhuber2015.pdf

Deep learning in neural networks

Independent component models have gained increasing interest in various fields of applications in recent years. The basic independent component model is a semiparametric model assuming that a p-variate observed random vector is a linear transformation of an unobserved vector of p independent latent variables. This linear transformation is given by an unknown mixing matrix, and one of the main objectives of independent component analysis (ICA) is to estimate an unmixing matrix by means of which the latent variables can be recovered. In this article, we discuss the basic independent component model in detail, define the concepts and analysis tools carefully, and consider two families of ICA estimates. The statistical properties (consistency, asymptotic normality, efficiency, robustness) of the estimates can be analyzed and compared via the so called gain matrices. Some extensions of the basic independent component model, such as models with additive noise or models with dependent observations, are briefly discussed. The article ends with a short example.


Keywords:

blind source separation;
fastICA;
independent component model;
independent subspace analysis;
mixing matrix;
overcomplete ICA;
undercomplete ICA;
unmixing matrix

Independent Component Analysis

Many problems in the sciences and engineering can be rephrased as optimization problems on matrix search spaces endowed with a so-called manifold structure. This book shows how to exploit the special structure of such problems to develop efficient numerical algorithms. It places careful emphasis on both the numerical formulation of the algorithm and its differential geometric abstraction--illustrating how good algorithms draw equally from the insights of differential geometry, optimization, and numerical analysis. Two more theoretical chapters provide readers with the background in differential geometry necessary to algorithmic development. In the other chapters, several well-known optimization methods such as steepest descent and conjugate gradients are generalized to abstract manifolds. The book provides a generic development of each of these methods, building upon the material of the geometric chapters. It then guides readers through the calculations that turn these geometrically formulated methods into concrete numerical algorithms. The state-of-the-art algorithms given as examples are competitive with the best existing algorithms for a selection of eigenspace problems in numerical linear algebra. Optimization Algorithms on Matrix Manifolds offers techniques with broad applications in linear algebra, signal processing, data mining, computer vision, and statistical analysis. It can serve as a graduate-level textbook and will be of interest to applied mathematicians, engineers, and computer scientists.

http://www.eeci-institute.eu/GSC2011/Photos-EECI/EECI-GSC-2011-M5/book_AMS.pdf

Optimization Algorithms on Matrix Manifolds

Independent Component Analysis.

This work investigates the problem of blind channel equalization in the presence of burst noise. Modeling impulsive noise distribution as generalized-Gaussian, we derive an appropriate criterion for blind equalization. Our method exploits Wijsman’s theorem and develops a maximal-invariant test of equally-likely pulsed signals against impulsive disturbance. We optimize the test and realize an adaptive equalizer, capable of not only mitigating intersymbol interference but also robust to impulsive disturbance. Numerical simulations, obtained on a baseband digital microwave radio system for amplitude-phase shift keying signaling in additive (generalized Gaussian and symmetric-alpha stable) impulsive environment, confirm the admissibility of the proposed equalizer in terms of robustness and steady convergence.

Adaptive Blind Equalization in Impulsive Noise

We derive a performance comparison between the conventional time-multiplexing training based scheme and the most-recently proposed data-dependent superimposed training scheme, when using linear receivers For both schemes, we derive an approximative closed-form expression for the Bit Error Rate (BER) and determine the optimal power allocation between pilot and data that minimizes the BER Simulations are conducted to assess the accuracy of the provided expressions and to determine contexts for which it is more interesting to opt for the data-dependent superimposed training scheme

http://wirelesslab.ca/File/pdf_files/conf/C141.pdf

Superimposed or time-multiplexed training: A performance comparison

This paper introduces new contrast functions for blind separation of sources with different time-frequency signatures. Two contrast functions based on the Kullback-Leibler and Jensen-Renyi divergences in the time-frequency (T-F) plane are introduced. Two iterative algorithms are proposed for the proposed contrasts optimization and source separation. One algorithm consists of spatial whitening and gradient-Jacobi maximization, combining Givens rotations and stochastic gradient. The second algorithm uses a quasi-Newton technique.

Contrast functions for blind source separation based on time-frequency information-theory

We combine the adaptive MMSE detector which employs an antenna array with interference cancellation in a multistage detection scheme. The proposed detectors are based on the signal subspace estimation. The dimension of the latter is decreased each time an interferer is removed from the received signal. The orthonormal projection approximation subspace tracking (OPAST) is used to adaptively estimate the proposed detectors and the spatial signatures. Compared with the MMSE detector, the resulting detectors offer a superior performance in terms of the BER.

https://ieeexplore.ieee.org/iel5/6675/17896/00831745.pdf

Subspace-based adaptive multiuser interference cancellation for CDMA system

The method involves equalizing a received signal (30) An interference induced by a source signal transmitted through channels other than a channel of interest, is estimated from the equalized signal in order to obtain a representative signal of the channel of interest An asymmetry code for the channels is estimated at random according to predetermined criteria during the estimation of the corresponding source signal An independent claim is also included for a code division multiple access (CDMA) signal receiver

Code division multiple access signal receiving method for universal mobile telecommunication system-frequency division duplex, involves estimating interference induced by source signal transmitted through channels

Karim Abed-Meraim

Papers

Adaptive Blind Equalization in Impulsive Noise

Superimposed or time-multiplexed training: A performance comparison

Contrast functions for blind source separation based on time-frequency information-theory

Subspace-based adaptive multiuser interference cancellation for CDMA system

Code division multiple access signal receiving method for universal mobile telecommunication system-frequency division duplex, involves estimating interference induced by source signal transmitted through channels