We introduce a novel adaptive approach for solving L1-minimization problems which frequently arise in compressed sensing. This is based on the recently introduced inverse scale space method. The scheme allows us to efficiently compute minimizers by solving a sequence of low dimensional nonegative least squares problems. This is joint work with Martin Burger, Michael Moeller and Martin Benning. Next we follow a discretization of this regularization path and apply it to sparse logistic regression. The method has computational advantages over the grid search method. This is joint work with Jianing Shi and Wotao Yin.
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