Submodular Point Processes with Applications to Machine Learning

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 themselves submodular (or supermodular). In this paper, we analyze the computational complexity of probabilistic inference in SPPs. We show that computing the partition function for SPPs (and Log-SPPs), requires exponential complexity in the worst case, and also provide algorithms which approximate SPPs up to polynomial factors. Moreover, for several subclasses of interesting submodular functions that occur in applications, we show how we can provide efficient closed form expressions for the partition functions, and thereby marginals and conditional distributions. We also show how SPPs are closed under mixtures, thus enabling maximum likelihood based strategies for learning mixtures of submodular functions. Finally, we argue how SPPs complement existing Log-SPP distributions, and are a natural model for several applications

SFO: A Toolbox for Submodular Function Optimization

In recent years, a fundamental problem structure has emerged as very useful in a variety of machine learning applications: Submodularity is an intuitive diminishing returns property, stating that adding an element to a smaller set helps more than adding it to a larger set. Similarly to convexity, submodularity allows one to efficiently find provably (near-) optimal solutions for large problems. We present SFO, a toolbox for use in MATLAB or Octave that implements algorithms for minimization and maximization of submodular functions. A tutorial script illustrates the application of submodularity to machine learning and AI problems such as feature selection, clustering, inference and optimized information gathering