Second-Order Separation of Multidimensional Sources with Constrained Mixing System

The case of sources that generate multidimensional signals, filling a subspace of dimensionality K, is considered. Different coordinate axes of the subspace (“subspace channels”) correspond to different signal portions generated by each source, e.g., data from different spectral bands or different modalities may be assigned to different subspace channels. The mixing system that generates observed signals from the underlying sources is modeled as superimposing within each subspace channel the contributions of the different sources. This mixing system is constrained as it allows no mixing of data that occurs in different subspace channels. An algorithm based on second order statistics is given which leads to a solution in closed form for the separating system. Correlations across different subspace channels are utilized by the algorithm, whereas properties such as higher-order statistics or spectral characteristics within subspace channels are not considered. A permutation problem of aligning different sources’ subspace channels is solved based on ordering of eigenvalues derived from the separating system. Effectiveness of the algorithm is demonstrated by application to multidimensional temporally i.i.d. Gaussian signals.