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 paper considers the problem of channel shortening and source separation in MIMO-OFDM systems. The basic OFDM system uses a guard interval to eliminate the inter symbol interference (ISI) and the intercarrier interference (ICI). These interferences persist when the length of the channel is superior to the GI. We need then to shorten the channel. Using MIMO-OFDM allows to increase the bit rate but suffers from the co-channel interference. In this paper, we are interested in the case where the number of sensors is larger than the number of sources. It is known that in this case, several time domain equalizers (TEQ) can be obtained to shorten the channel. We exploit then these freedom degrees to suppress the co-channel interference and to control the Target Impulse Response (TIR) quality. Our processed system can be viewed as in form of N t parallel SISO-OFDM systems where N t is the number of sources. Simulation results are provided to illustrate this claim and to assess the performance of the proposed method.

Combined channel shortening and source separation with TIR controlled quality in MIMO-OFDM systems

It is well known that multipath channel and carrier frequency offset destroy the orthogonality between subcarriers in the downlink of MC-CDMA and degrade the system performance In this paper, a new low complexity algorithm is proposed to iteratively estimate the carrier frequency offset (CFO) and the channel equalizer using only the guard interval (GI) redundancy, or excess codes (EC) or both (GI) and (EC) without additional pilots Simulations illustrate and compare the performance efficiency of these schemes in different scenarios

Karim Abed-Meraim

Papers

Combined channel shortening and source separation with TIR controlled quality in MIMO-OFDM systems

Blind Channel Equalization and Carrier Frequency Offset Estimation for MC-CDMA Systems using Guard Interval Redundancy and Excess Codes