  We study the convergence rate of the proximal-gradient homotopy algorithm
applied to norm-regularized linear least squares problems, for a general class
of norms. The homotopy algorithm reduces the regularization parameter in a
series of steps, and uses a proximal-gradient algorithm to solve the problem at
each step. Proximal-gradient algorithm has a linear rate of convergence given
that the objective function is strongly convex, and the gradient of the smooth
component of the objective function is Lipschitz continuous. In many
applications, the objective function in this type of problem is not strongly
convex, especially when the problem is high-dimensional and regularizers are
chosen that induce sparsity or low-dimensionality. We show that if the linear
sampling matrix satisfies certain assumptions and the regularizing norm is
decomposable, proximal-gradient homotopy algorithm converges with a
\emph{linear rate} even though the objective function is not strongly convex.
Our result generalizes results on the linear convergence of homotopy algorithm
for $l_1$-regularized least squares problems. Numerical experiments are
presented that support the theoretical convergence rate analysis.
