Frederik Van Eeghem

Blind Multichannel Deconvolution and Convolutive Extensions of Canonical Polyadic and Block Term Decompositions

Tensor decompositions such as the canonical polyadic decomposition (CPD) or the block term decomposition (BTD) are basic tools for blind signal separation. Most of the literature concerns instantaneous mixtures/memoryless channels. In this paper, we focus on convolutive extensions. More precisely, we present a connection between convolutive CPD/BTD models and coupled but instantaneous CPD/BTD. We derive a new identifiability condition dedicated to convolutive low-rank factorization problems. We explain that under this condition, the convolutive extension of CPD/BTD can be computed by means of an algebraic method, guaranteeing perfect source separation in the noiseless case. In the inexact case, the algorithm can be used as a cheap initialization for an optimization-based method. We explain that, in contrast to the memoryless case, convolutive signal separation is in certain cases possible despite only two-way diversities (e.g., space

Tensor Decompositions With Several Block-Hankel Factors and Application in Blind System Identification

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Second-order tensor-based convolutive ICA: Deconvolution versus tensorization

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Coupled and incomplete tensors in blind system identification

Several applications in biomedical data processing, telecommunications, or chemometrics can be tackled by computing a structured tensor decomposition. In this paper, we focus on tensor decompositions with two or more block-Hankel factors, which arise in blind multiple-input-multiple-output (MIMO) convolutive system identification. By assuming statistically independent inputs, the blind system identification problem can be reformulated as a Hankel structured tensor decomposition. By capitalizing on the available block-Hankel and tensorial structure, a relaxed uniqueness condition for this structured decomposition is obtained. This condition is easy to check, yet very powerful. The uniqueness condition also forms the basis for two subspace-based algorithms, able to blindly identify linear underdetermined MIMO systems with finite impulse response.

Algorithms for Canonical Polyadic Decomposition With Block-Circulant Factors

Independent component analysis (ICA) research has been driven by various applications in biomedical signal separation, telecommunications, speech analysis, and more. One particular class of algorithms for instantaneous ICA uses tensors, which have useful properties. In an attempt to port these properties to convolutive methods, we zoom in on an existing method that uses second-order statistics. By pointing out links in the literature, we show that this method is in fact a typical tensor-based method, even though this was not recognized by the authors at the time. The existing method mentioned above can be interpreted as a tensorization step followed by a deconvolution step. However, as sometimes done in literature, one may consider using the opposite approach; starting with a deconvolution step and then tensorizing the remaining instantaneous mixture. Because subspace-based deconvolution can be slow, we propose a fast variant which uses only partial information. We then use this variant to compare the approach starting with tensorization and the one starting with deconvolution.