  Two of the most fundamental prototypes of greedy optimization are the
matching pursuit and Frank-Wolfe algorithms. In this paper, we take a unified
view on both classes of methods, leading to the first explicit convergence
rates of matching pursuit methods in an optimization sense, for general sets of
atoms. We derive sublinear ($1/t$) convergence for both classes on general
smooth objectives, and linear convergence on strongly convex objectives, as
well as a clear correspondence of algorithm variants. Our presented algorithms
and rates are affine invariant, and do not need any incoherence or sparsity
assumptions.
