  We consider the well-studied problem of decomposing a vector time series
signal into components with different characteristics, such as smooth,
periodic, nonnegative, or sparse. We describe a simple and general framework in
which the components are defined by loss functions (which include constraints),
and the signal decomposition is carried out by minimizing the sum of losses of
the components (subject to the constraints). When each loss function is the
negative log-likelihood of a density for the signal component, this framework
coincides with maximum a posteriori probability (MAP) estimation; but it also
includes many other interesting cases. Summarizing and clarifying prior
results, we give two distributed optimization methods for computing the
decomposition, which find the optimal decomposition when the component class
loss functions are convex, and are good heuristics when they are not. Both
methods require only the masked proximal operator of each of the component loss
functions, a generalization of the well-known proximal operator that handles
missing entries in its argument. Both methods are distributed, i.e., handle
each component separately. We derive tractable methods for evaluating the
masked proximal operators of some loss functions that, to our knowledge, have
not appeared in the literature.
