Lasso, fractional norm and structured sparse estimation using a Hadamard product parametrization

Using a multiplicative reparametrization, it is shown that a subclass of Lq penalties with q less than or equal to one can be expressed as sums of L2 penalties. It follows that the lasso and other norm-penalized regression estimates may be obtained using a very simple and intuitive alternating ridge regression algorithm. As compared to a similarly intuitive EM algorithm for Lq optimization, the proposed algorithm avoids some numerical instability issues and is also competitive in terms of speed. Furthermore, the proposed algorithm can be extended to accommodate sparse high-dimensional scenarios, generalized linear models, and can be used to create structured sparsity via penalties derived from covariance models for the parameters. Such model-based penalties may be useful for sparse estimation of spatially or temporally structured parameters.