  Continuous submodular functions are a category of generally
non-convex/non-concave functions with a wide spectrum of applications. The
celebrated property of this class of functions - continuous submodularity -
enables both exact minimization and approximate maximization in poly. time.
Continuous submodularity is obtained by generalizing the notion of
submodularity from discrete domains to continuous domains. It intuitively
captures a repulsive effect amongst different dimensions of the defined
multivariate function.
  In this paper, we systematically study continuous submodularity and a class
of non-convex optimization problems: continuous submodular function
maximization. We start by a thorough characterization of the class of
continuous submodular functions, and show that continuous submodularity is
equivalent to a weak version of the diminishing returns (DR) property. Thus we
also derive a subclass of continuous submodular functions, termed continuous
DR-submodular functions, which enjoys the full DR property. Then we present
operations that preserve continuous (DR-)submodularity, thus yielding general
rules for composing new submodular functions. We establish intriguing
properties for the problem of constrained DR-submodular maximization, such as
the local-global relation. We identify several applications of continuous
submodular optimization, ranging from influence maximization, MAP inference for
DPPs to provable mean field inference. For these applications, continuous
submodularity formalizes valuable domain knowledge relevant for optimizing this
class of objectives. We present inapproximability results and provable
algorithms for two problem settings: constrained monotone DR-submodular
maximization and constrained non-monotone DR-submodular maximization. Finally,
we extensively evaluate the effectiveness of the proposed algorithms.
