Linear mixture model

In a general linear mixture model, the measurement random vector $\textbf{y} \in \mathbb{R}^{m\times 1}$ can be described as

$$\textbf{y} = \textbf{A}\textbf{s} + \textbf{n}$$ (2)

where $\textbf{s} \in \mathbb{R}^{n\times 1}$ is an independent random vector representing the sources, $\textbf{A} \in \mathbb{R}^{m\times n}$ is the unknown mixing matrix, $\textbf{n} \in \mathbb{R}^{m\times 1}$ is an independent random vector with Gaussian white noise representing sensor noise.

For $m \geq n$, several algorithms exist that estimate the unmixing matrix $\textbf{W} = \textbf{A}^{-1}$ sufficiently well [4]. For $m < n$ the mixing matrix is not square and the problem cannot be solved without additional information about the sources. In the absence of noise, if only source $i$ is active, the output signal $\textbf{y}$ will be aligned with the vector representing the $i$-th column of $\textbf{A}$, the $i$-th ``basis vector'' [7]. Therefore, if the sources are sparse according to the model described in (1), most of the output samples $\textbf{y}$ will be aligned with one of the basis vectors (see Fig. 2(a)).

Using this geometrical insight a large number of estimators for the mixing matrix have been proposed, amongst them a technique using overcomplete representations [6], a line spectrum estimation method [14] and a number of geometric algorithms [15,16]. Once the mixing matrix has been estimated, the original sources can be recuperated with the shortest-path algorithm introduced in [7].