Signal Separation For Equal

We introduce undercomplete independent component analysis (uICA), a method for extracting K signals from M K mixtures of N M source signals. The mixtures x = (x 1 ; : : :; x M) T are formed from a linear combination of the independent source signals s = (s 1 ; : : :; s N) T using an M N mixing matrix A, so that x = As. For the case N = K = M, Bell and Sejnowski Bell and Sejnowski, 1995] showed that a square N N unmixing matrix W can be found by maximising the joint entropy of M signals (Y 1 ; : : :; Y N) T = (Wx), where is a monotonic, non-linear function. Using a similar approach, we show that a K M unmix-ing matrix W can be used to recover K M signals (s 1 ; : : :; s K) T by maximising the joint entropy of K signals Y = (Wx). The matrix W is essentially a pseudo-inverse of the M N mixing matrix A. A diierent and widely used method for reducing the size of W is to perform principal component analysis (PCA) on the data set, and to use only the L principal components with the largest eigenvalues as input to ICA. This results in an L L unmixing matrix W. However, there is no a priori reason to assume that independent components exist in only the L-D subspace deened by the L principal components with largest eigenvectors. Thus, discarding some eigenvectors may also corrupt or discard independent components. In contrast, uICA does not discard any independent components in the data set, and can extract between 1 and M signals from x. The method is demonstrated on mixtures of high kurtosis (speech and music) and Gaussian signals.