Discrete Signal Processing with Set Functions

Set functions are functions (or signals) indexed by the power set (set of all subsets) of a finite set $N$. They are ubiquitous in many application domains. For example, they are equivalent to node- or edge-weighted hypergraphs and to cooperative games in game theory. Further, the subclass of submodular functions occurs in many optimization and machine learning problems. In this paper, we derive discrete-set signal processing (SP), a shift-invariant linear signal processing framework for set functions. Discrete-set SP provides suitable definitions of shift, shift-invariant systems, convolution, Fourier transform, frequency response, and other SP concepts. Different variants are possible due to different possible shifts. Discrete-set SP is inherently different from graph SP as it distinguishes the neighbors of an index $A\subseteq N$, i.e., those with one elements more or less by providing $n = |N|$ shifts. Finally, we show three prototypical applications and experiments with discrete-set SP including compression in submodular function optimization, sampling for preference elicitation in auctions, and novel power set neural networks.