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We study an optimization program over nonnegative Borel measures thatencourages sparsity in its solution. Efficient solvers for this program are inincreasing demand, as it arises when learning from data generated by a`continuum-of-subspaces' model, a recent trend with applications in signalprocessing, machine learning, and high-dimensional statistics. We prove thatthe conditional gradient method (CGM) applied to this infinite-dimensionalprogram, as proposed recently in the literature, is equivalent to the exchangemethod (EM) applied to its Lagrangian dual, which is a semi-infinite program.In doing so, we formally connect such infinite-dimensional programs to thewell-established field of semi-infinite programming.  On the one hand, the equivalence established in this paper allows us toprovide a rate of convergence for EM which is more general than those existingin the literature. On the other hand, this connection and the resultinggeometric insights might in the future lead to the design of improved variantsof CGM for infinite-dimensional programs, which has been an active researchtopic. CGM is also known as the Frank-Wolfe algorithm.

Sparse Inverse Problems Over Measures: Equivalence of the Conditional Gradient and Exchange Methods