  We study the convergence properties of a general inertial first-order
proximal splitting algorithm for solving nonconvex nonsmooth optimization
problems. Using the Kurdyka--\L ojaziewicz (KL) inequality we establish new
convergence rates which apply to several inertial algorithms in the literature.
Our basic assumption is that the objective function is semialgebraic, which
lends our results broad applicability in the fields of signal processing and
machine learning. The convergence rates depend on the exponent of the
"desingularizing function" arising in the KL inequality. Depending on this
exponent, convergence may be finite, linear, or sublinear and of the form
$O(k^{-p})$ for $p>1$.
