  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 $\ell_q$ penalties is challenging. Faster algorithms for
computing solution paths for these penalties are needed because $\ell_q$
penalized regression problems can be nonconvex and especially burdensome to
solve. In this paper, we show that a reparameterization of $\ell_q$ penalized
regression problems is more amenable to pathwise coordinate descent algorithms.
This allows us to improve computation of the mode-thresholding function for
$\ell_q$ 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.
