We introduce a class of discrete divergences on sets (equivalently binary vectors) that we call the submodular-Bregman divergences. We consider two kinds, defined either from tight modular upper or tight modular lower bounds of a submodular function. We show that the properties of these divergences are analogous to the (standard continuous) Bregman divergence. We demonstrate how they generalize...

Can we summarize multi-category data based on user preferences in a scalable manner? Many utility functions used for data summarization satisfy submodularity, a natural diminishing returns property. We cast personalized data summarization as an instance of a general submodular maximization problem subject to multiple constraints. We develop the first practical and FAst coNsTrained submOdular Ma...

We introduce a class of discrete point processes that we call the Submodular Point Processes (SPPs). These processes are characterized via a submodular (or supermodular) function, and naturally model notions of information, coverage and diversity, as well as cooperation. Unlike Log-submodular and Log-supermodular distributions (Log-SPPs) such as determinantal point processes (DPPs), SPPs are th...

The Cheeger inequality for undirected graphs, which relates the conductance of an undirected graph and the second smallest eigenvalue of its normalized Laplacian, is a cornerstone of spectral graph theory. The Cheeger inequality has been extended to directed graphs and hypergraphs using normalized Laplacians for those, that are no longer linear but piecewise linear transformations. In this pape...

We consider the problem of minimizing a function represented as a sum of submodular terms. We assume each term allows an efficient computation of exchange capacities. This holds, for example, for terms depending on a small number of variables, or for certain cardinalitydependent terms. A naive application of submodular minimization algorithms would not exploit the existence of specialized excha...

We present a practical and powerful new framework for both unconstrained and constrained submodular function optimization based on discrete semidifferentials (suband super-differentials). The resulting algorithms, which repeatedly compute and then efficiently optimize submodular semigradients, offer new and generalize many old methods for submodular optimization. Our approach, moreover, takes s...

Motivated by economic thought, a recent research agenda has suggested the algorithmic study of combinatorial optimization problems under functions which satisfy the property of decreasing marginal cost. A natural first step to model such functions is to consider submodular functions. However, many fundamental problems have turned out to be extremely hard to approximate under general submodular ...

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...