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

Using a multiplicative reparametrization, I show that a subclass of $L_q$ penalties with $q\leq 1$ can be expressed as sums of $L_2$ 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 $L_q$ 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.