  We present a forward-backward-based algorithm to minimize a sum of a
differentiable function and a nonsmooth function, both being possibly
nonconvex. The main contribution of this work is to consider the challenging
case where the nonsmooth function corresponds to a sum of non-convex functions,
resulting from composition between a strictly increasing, concave,
differentiable function and a convex nonsmooth function. The proposed variable
metric Composite Function Forward-Backward algorithm (C2FB) circumvents the
explicit, and often challenging, computation of the proximity operator of the
composite functions through a majorize-minimize approach. Precisely, each
composite function is majorized using a linear approximation of the
differentiable function, which allows one to apply the proximity step only to
the sum of the nonsmooth functions. We prove the convergence of the algorithm
iterates to a critical point of the objective function leveraging the
Kurdyka-\L ojasiewicz inequality. The convergence is guaranteed even if the
proximity operators are computed inexactly, considering relative errors. We
show that the proposed approach is a generalization of reweighting methods,
with convergence guarantees. In particular, applied to the log-sum function,
our algorithm reduces to a generalized version of the celebrated reweighted
$\ell_1$ method. Finally, we show through simulations on an image processing
problem that the proposed C2FB algorithm necessitates less iterations to
converge and leads to better critical points compared with traditional
reweighting methods and classic forward-backward algorithms.
