Alternating Group Lasso for Block-Term Tensor Decomposition and Application to ECG Source Separation

In some applications, blind source separation can be performed by computing an approximate block-term tensor decomposition (BTD), under much milder constraints than matrix-based techniques. However, choosing the BTD model structure (i.e., the number of blocks and their ranks) is a difficult problem, and the standard least-squares formulation can be ill-posed. This paper proposes an alternating group lasso algorithm to compute approximate low-rank BTDs. It solves, in a provably convergent manner, a well-posed mixed-norm regularized tensor approximation problem that allows estimating the model parameters and its structure jointly. A variant is also put forward for dealing with linearly constrained blocks, motivated by the problem of blind separation of sums of complex exponentials, which can be cast as a low-rank Hankel-structured block-term tensor approximation problem. An experimental comparison with a standard nonlinear least-squares algorithm on synthetic tensor data indicates that the proposed algorithm is much more robust with respect to initialization. We also apply the constrained variant to the extraction of atrial activity from semi-synthetic and real-world electrocardiogram recordings during atrial fibrillation episodes. Our results show its ability to consistently select an adequate structure and to extract multiple signals which can be physiologically interpreted as atrial fibrillation patterns.