  In this paper, we introduce TITAN, a novel inerTIal block majorizaTion
minimizAtioN framework for non-smooth non-convex optimization problems. To the
best of our knowledge, TITAN is the first framework of block-coordinate update
method that relies on the majorization-minimization framework while embedding
inertial force to each step of the block updates. The inertial force is
obtained via an extrapolation operator that subsumes heavy-ball and
Nesterov-type accelerations for block proximal gradient methods as special
cases. By choosing various surrogate functions, such as proximal, Lipschitz
gradient, Bregman, quadratic, and composite surrogate functions, and by varying
the extrapolation operator, TITAN produces a rich set of inertial
block-coordinate update methods. We study sub-sequential convergence as well as
global convergence for the generated sequence of TITAN. We illustrate the
effectiveness of TITAN on two important machine learning problems, namely
sparse non-negative matrix factorization and matrix completion.
