Submodularity is an important  property of set functions with deep theoretical results  and various  applications. Submodular systems appear in many applicable area, for example machine learning, economics, computer vision, social science, game theory and combinatorial optimization.  Nowadays submodular functions optimization has been attracted by many researchers.  Pendant pairs of a symmetric...

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 Lovász 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 analy...

A number of discrete and continuous optimization problems in machine learning are related to convex minimization problems under submodular constraints. In this paper, we deal with a submodular function with a directed graph structure, and we show that a wide range of convex optimization problems under submodular constraints can be solved much more efficiently than general submodular optimizatio...

In this paper we develop a framework of submodular optimization algorithms in line with the mirror-descent style of algorithms for convex optimization. We use the fact that a submodular function has both a subdifferential and a superdifferential, which enables us to formulate algorithms for both submodular minimization and maximization. This reveals a unifying framework for a number of submodul...

We investigate a new optimization problem involving minimizing the Ratio of two Submodular (RS) functions. We argue that this problem occurs naturally in several real world applications. We then show the connection between this problem and several related problems including minimizing the difference between submodular functions (Iyer & Bilmes, 2012b; Narasimhan & Bilmes, 2005), and to submodula...

Submodular functions are discrete functions that model laws of diminishing returns and enjoy numerous algorithmic applications. They have been used in many areas, including combinatorial optimization, machine learning, and economics. In this work we study submodular functions from a learning theoretic angle. We provide algorithms for learning submodular functions, as well as lower bounds on the...

Motivation: Submodular optimization, a discrete analogue to continuous convex optimization, has been used with great success in many fields but is not yet widely used in biology. We apply submodular optimization to the problem of removing redundancy in protein sequence data sets. This is a common step in many bioinformatics and structural biology workflows, including creation of non-redundant t...

The scalability of submodular optimization methods is critical for their usability in practice. In this paper, we study the reducibility of submodular functions, a property that enables us to reduce the solution space of submodular optimization problems without performance loss. We introduce the concept of reducibility using marginal gains. Then we show that by adding perturbation, we can endow...