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Pathwise coordinate descent algorithms have been used to compute entire solution paths for lasso and other penalized regression problems quickly with great success. They improve upon cold start algorithms by solving the problems that make up the solution path sequentially for an ordered set of tuning parameter values, instead of solving each problem separately. However, extending pathwise coordinate descent algorithms to more the general bridge or power family of penalties is challenging. Faster algorithms for computing solution paths for these penalties are needed because these penalized regression problems can be nonconvex and especially burdensome to solve. In this article, we show that a reparameterization of these penalized regression problems is more amenable to pathwise coordinate descent algorithms. This allows us to improve computation of the mode-thresholding function for penalized regression problems in practice and introduce two separate pathwise algorithms. We show that either pathwise algorithm is faster than the corresponding cold start alternative, and demonstrate that different pathwise algorithms may be more likely to reach better solutions. Supplemental materials for this article are available online. 

Lightning is a destructive and highly visible product of severe storms, yet there is still much to be learned about the conditions under which lightning is most likely to occur. The GOES-16 and GOES-17 satellites, launched in 2016 and 2018 by NOAA and NASA, collect a wealth of data regarding individual lightning strike occurrence and potentially related atmospheric variables. The acute nature and inherent spatial correlation in lightning data renders standard regression analyses inappropriate. Further, computational considerations are foregrounded by the desire to analyze the immense and rapidly increasing volume of lightning data. We present a new computationally feasible method that combines spectral and Laplace approximations in an EM algorithm, denoted SLEM, to fit the widely popular log-Gaussian Cox process model to large spatial point pattern datasets. In simulations we find SLEM is competitive with contemporary techniques in terms of speed and accuracy. When applied to two lightning datasets, SLEM provides better out-of-sample prediction scores and quicker runtimes, suggesting its particular usefulness for analyzing lightning data which tend to have sparse signals.