On Some New Ideas and Algorithms for Independent Component Analysis (Nonlinear Analysis and Convex Analysis)
Pando Georgiev, Andrzej S Cichocki · Institutional Repositories DataBase (IRDB) · 2002
For every integer $p\geq 4$ , even, we consider aspecific optimization problem $\mathrm{O}\mathrm{P}(\mathrm{p})$ arising from the blind source extraction problem (BSE) and prove that every local maximum of $\mathrm{O}\mathrm{P}(\mathrm{p})$ is asolution of (BSE) in sense that it extracts one source signal from alinear mixture of unknown statistically independent signals.We construct an algorithm for solving $\mathrm{O}\mathrm{P}(\mathrm{p})$ with rate of convergence $p-1$ .We propose new sufficient conditions for separation of source signals, stating that the separation is possible, if the source signals have different autocorrelation or cumulant functions (depending on time delay).We show that the problem of blind source separation of signals can be qonverted to asymmetric eigenvalue problem of ageneralized cumulant matrices if these matrices have distinct eigenvalues.We propose new algorithms, based on non-smooth analysis and optimization theory, which disperse the eigenvalues of these generalized cumulant matrices.1IntroductionThe problem of independent component analysis is formulated as follows: we observe sensor signals (random variables) $\mathrm{x}(t)=[x_{1}(t), \ldots, x_{m}(t)]^{T}$ and want to represent them as linear mixture of random variables $\mathrm{s}(t)=[s_{1}(t), \ldots, s_{n}(t)]^{T}$ , which are independent, as much as possible:where Ais $n\cross n$ non-singular matrix.The problem of blind source extraction (BSE), which we shall consider, is formulated as follows: for given source signals (random variables) $\mathrm{x}(t)=[x_{1}(t), \ldots, x_{m}(t)]^{T}$ and knowing that they are obtained as alinear mixture (1), the task is to find $\mathrm{s}(t)$ and the matrix A. In general this is impossible, but if $s_{i}$ , $i=1$ , $\ldots$ , $n$ are statistically independent and Ais nonsingular, then this is possible up to permutation and scaling, i.e. we can obtain ADP, where $\mathrm{D}$ and $\mathrm{P}$ are unknown diagonal and permutation matrices respectively.Therefore, we can obtain $d_{i}s_{p}.\cdot(t)$, where $p_{i}$ is a permutation of $\{$ 1, $\ldots$ , $n\}$ (unknown) and $d_{i}$ are scaling coefficients (unknown).The literature about independent component analysis and BSE problem is huge (see for instance [16] and references therein).