Discrete Signal Processing with Set Functions
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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⊆ 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.
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