  Submodular functions are relevant to machine learning for at least two
reasons: (1) some problems may be expressed directly as the optimization of
submodular functions and (2) the lovasz extension of submodular functions
provides a useful set of regularization functions for supervised and
unsupervised learning. In this monograph, we present the theory of submodular
functions from a convex analysis perspective, presenting tight links between
certain polyhedra, combinatorial optimization and convex optimization problems.
In particular, we show how submodular function minimization is equivalent to
solving a wide variety of convex optimization problems. This allows the
derivation of new efficient algorithms for approximate and exact submodular
function minimization with theoretical guarantees and good practical
performance. By listing many examples of submodular functions, we review
various applications to machine learning, such as clustering, experimental
design, sensor placement, graphical model structure learning or subset
selection, as well as a family of structured sparsity-inducing norms that can
be derived and used from submodular functions.
