  Submodular functions describe a variety of discrete problems in machine
learning, signal processing, and computer vision. However, minimizing
submodular functions poses a number of algorithmic challenges. Recent work
introduced an easy-to-use, parallelizable algorithm for minimizing submodular
functions that decompose as the sum of "simple" submodular functions.
Empirically, this algorithm performs extremely well, but no theoretical
analysis was given. In this paper, we show that the algorithm converges
linearly, and we provide upper and lower bounds on the rate of convergence. Our
proof relies on the geometry of submodular polyhedra and draws on results from
spectral graph theory.
