Using Atomic Bounds to Get Sub-modular Approximations

Set functions and inferences over them play an important role in many AI-related problems. They are commonly used in kernel-based methods, game theory, uncertainty representation and preference modelling. Of particular importance are additive set-functions and their associated expectation operator, as well as sub-modular functions. However, specifying precisely and completely such set functions can be a daunting task as well as a lot of data and knowledge. When time or information is missing, it may be handy to have approximating tools of such set-functions, that require less information and possibly enjoy nice mathematical properties. In this paper, we show that if the only information we have are atomic bounds of such functions, we can build conservative approximations that are either sub- or super-modular. We then illustrate the potential use of our approximation on a convolution-based signal processing problem.