  In recent years, there has been a growing interest in mathematical models
leading to the minimization, in a symmetric matrix space, of a Bregman
divergence coupled with a regularization term. We address problems of this type
within a general framework where the regularization term is split in two parts,
one being a spectral function while the other is arbitrary. A Douglas-Rachford
approach is proposed to address such problems and a list of proximity operators
is provided allowing us to consider various choices for the fit-to-data
functional and for the regularization term. Numerical experiments show the
validity of this approach for solving convex optimization problems encountered
in the context of sparse covariance matrix estimation. Based on our theoretical
results, an algorithm is also proposed for noisy graphical lasso where a
precision matrix has to be estimated in the presence of noise. The nonconvexity
of the resulting objective function is dealt with a majorization-minimization
approach, i.e. by building a sequence of convex surrogates and solving the
inner optimization subproblems via the aforementioned Douglas-Rachford
procedure. We establish conditions for the convergence of this iterative scheme
and we illustrate its good numerical performance with respect to
state-of-the-art approaches.
