We give under weak assumptions a complete combinatorial characterization of identifiability for linear mixtures of finite alphabet sources, with unknown mixing weights and unknown source signals, but known alphabet. This is based on a detailed treatment of the case of a single linear mixture. Notably, our identifiability analysis applies also to the case of unknown number of sources. We provide sufficient and necessary conditions for identifiability and give a simple sufficient criterion together with an explicit construction to determine the weights and the source signals for deterministic data by taking advantage of the hierarchical structure within the possible mixture values. We show that the probability of identifiability is related to the distribution of a hitting time and converges exponentially fast to one when the underlying sources come from a discrete Markov process. Finally, we explore our theoretical results in a simulation study. This paper extends and clarifies the scope of scenarios for which blind source separation becomes meaningful.
There are further variations of the BSS problem. Some of them are associated with Independent Component Analysis (ICA) (see e.g., @cite_15 ), which is based on the stochastic independence of the different sources (assumed to be random). ICA can be a powerful tool for (over)determined models ( @math ) @cite_15 and there are approaches for underdetermined multiple linear mixture models ( @math ) as well @cite_21 . However, ICA is not applicable for single linear mixtures ( @math ), as the error terms of the single sources sum up to a single error term such that stochastic independence of the sources becomes irrelevant.
@cite_15 @cite_21
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This paper proposes to generalize the notion of independent component analysis (ICA) to the notion of multidimensional independent component analysis (MICA). We start from the ICA or blind source separation (BSS) model and show that it can be uniquely identified provided it is properly parameterized in terms of one-dimensional subspaces. From this standpoint, the BSS ICA model is generalized to multidimensional components. We discuss how ICA standard algorithms can be adapted to MICA decomposition. The relevance of these ideas is illustrated by a MICA decomposition of ECG signals. Blind signal separation (BSS) and independent component analysis (ICA) are emerging techniques of array processing and data analysis that aim to recover unobserved signals or "sources" from observed mixtures (typically, the output of an array of sensors), exploiting only the assumption of mutual independence between the signals. The weakness of the assumptions makes it a powerful approach, but it requires us to venture beyond familiar second order statistics, The objectives of this paper are to review some of the approaches that have been developed to address this problem, to illustrate how they stem from basic principles, and to show how they relate to each other. Independent component analysis (ICA) is a computational method for separating a multivariate signal into subcomponents assuming the mutual statistical independence of the non-Gaussian source signals. The classical independent components analysis (ICA) framework usually assumes linear combinations of independent sources over the field of real-valued numbers R. In this paper, we investigate binary ICA for or mixtures (bICA), which can find applications in many domains including medical diagnosis, multi-cluster assignment, Internet tomography and network resource management. We prove that bICA is uniquely identifiable under the disjunctive generation model, and propose a deterministic iterative algorithm to determine the distribution of the latent random variables and the mixing matrix. The inverse problem to infer the values of latent variables is also considered for noisy measurements. We conduct an extensive simulation study to verify the effectiveness of the propose algorithm and present examples of real-world applications where bICA can be applied. In this paper, a novel algorithm to blindly separate an instantaneous linear underdetermined mixture of nonstationary sources is proposed. It means that the number of sources exceeds the number of channels of the available data. The separation is based on the working assumption that the sources are piecewise stationary with a different variance in each block. It proceeds in two steps: 1) estimating the mixing matrix, and 2) computing the optimum beamformer in each block to maximize the signal-to-interference ratio of each separated signal with respect to the remaining signals. Estimating the mixing matrix is accomplished through a specialized tensor decomposition of the set of sample covariance matrices of the received mixture in each block. It utilizes optimum weighting, which allows statistically efficient (CRB attaining) estimation provided that the data obey the assumed Gaussian piecewise stationary model. In simulations, performance of the algorithm is successfully tested on blind separation of 16 speech signals from nine linear instantaneous mixtures of these signals.