Tensor-Based Independent Component Analysis: from Instantaneous to Convolutive Mixtures

Data is everywhere, but it is only useful if information can be extracted from it. To this end, data analysis techniques have been developed to find features underlying the data. One class of data analysis techniques uses tensor decompositions. Tensors are higher-dimensional extensions of vectors and matrices and allow us to represent multiway data in a natural way. One of the key strengths of tensors is that their decompositions are unique under mild conditions without imposing additional constraints, unlike matrix decompositions. This property opens up many interesting applications of tensor decompositions, in particular with respect to blind source separation and independent component analysis. Independent component analysis (ICA) tries to find the statistically independent signals underlying a mixture, which is useful in many fields such as telecommunications, speech separation and biomedical data analysis. These mixtures are often modeled as instantaneous mixtures of source signals and tensor methods that can blindly separate such mixtures are well known. However, in many applications a convolutive mixture model is more appropriate to take delays and reflections of the source signals into account. This is for instance the case for speech signals in a room or telecommunications signals impinging on antennas. When the mixtures are convolutive, additional structure arises in the separation problem. Current tensor-based methods in the literature do not fully exploit this Toeplitz or Hankel structure. This thesis presents new subspace-based methods that take more of this structure into account, which leads to efficient and more accurate results than the current state-of-the-art tensor methods. Additionally, relaxed uniqueness bounds are formulated that exploit the available structure as well. This thesis also presents a method for a related structured tensor decomposition in which all factor matrices have block-circulant structure. Apart from exploiting the structure that appears in convolutive ICA, this thesis also presents how known techniques in instantaneous ICA can be ported to the convolutive case. This includes coupled tensor decompositions to combine second- and fourth-order statistics, and the usage of incomplete tensors to reduce the computational complexity. Another interesting question unrelated to convolutive ICA is whether we can compare underlying tensor factors without having to compute their full decompositions. This is highly relevant for any tensor classification problem but has only received limited attention in research. In this thesis, we present foundational theorems and algorithms that show how tensor factors can be compared in two modes for several underlying tensor decompositions. One of these methods is used further to develop an algorithm that is able to compare the cluster centers in two different datasets, without having to compute the actual clusters.

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