We address the problem of image collection summarization by learning mixtures of submodular functions. Submodularity is useful for this problem since it naturally represents characteristics such as fidelity and diversity, desirable for any summary. Several previously proposed image summarization scoring methodologies, in fact, instinctively arrived at submodularity. We provide classes of submod...

We investigate two new optimization problems — minimizing a submodular function subject to a submodular lower bound constraint (submodular cover) and maximizing a submodular function subject to a submodular upper bound constraint (submodular knapsack). We are motivated by a number of real-world applications in machine learning including sensor placement and data subset selection, which require ...

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 ...

Many computer vision problems require optimization of binary non-submodular energies. We propose a general optimization framework based on local submodular approximations (LSA). Unlike standard LP relaxation methods that linearize the whole energy globally, our approach iteratively approximates the energies locally. On the other hand, unlike standard local optimization methods (e.g. gradient de...

Submodular functions arewell-studied in combinatorial optimization, game theory and economics. The natural diminishing returns property makes them suitable for many applications. We study an extension of monotone submodular functions, which we call weakly submodular functions. Our extension is somewhat unusual in that it includes some (mildly) supermodular functions. We show that several natura...

We design a class of submodular functions meant for document summarization tasks. These functions each combine two terms, one which encourages the summary to be representative of the corpus, and the other which positively rewards diversity. Critically, our functions are monotone nondecreasing and submodular, which means that an efficient scalable greedy optimization scheme has a constant factor...

We introduce a method to learn a mixture of submodular “shells” in a large-margin setting. A submodular shell is an abstract submodular function that can be instantiated with a ground set and a set of parameters to produce a submodular function. A mixture of such shells can then also be so instantiated to produce a more complex submodular function. What our algorithm learns are the mixture weig...