  This technical note considers the problems of blind sparse learning and
inference of electrogram (EGM) signals under atrial fibrillation (AF)
conditions. First of all we introduce a mathematical model for the observed
signals that takes into account the multiple foci typically appearing inside
the heart during AF. Then we propose a reconstruction model based on a fixed
dictionary and discuss several alternatives for choosing the dictionary. In
order to obtain a sparse solution that takes into account the biological
restrictions of the problem, a first alternative is using LASSO regularization
followed by a post-processing stage that removes low amplitude coefficients
violating the refractory period characteristic of cardiac cells. As an
alternative we propose a novel regularization term, called cross products LASSO
(CP-LASSO), that is able to incorporate the biological constraints directly
into the optimization problem. Unfortunately, the resulting problem is
non-convex, but we show how it can be solved efficiently in an approximated way
making use of successive convex approximations (SCA). Finally, spectral
analysis is performed on the clean activation sequence obtained from the sparse
learning stage in order to estimate the number of latent foci and their
frequencies. Simulations on synthetic and real data are provided to validate
the proposed approach.
