Discrete Signal Processing on Meet/Join Lattices
Autor: | Püschel, Markus, Seifert, Bastian, Wendler, Chris |
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Rok vydání: | 2020 |
Předmět: | |
Zdroj: | IEEE Transactions on Signal Processing, Vol. 69, pp. 3571-3584, 2021 |
Druh dokumentu: | Working Paper |
DOI: | 10.1109/TSP.2021.3081036 |
Popis: | A lattice is a partially ordered set supporting a meet (or join) operation that returns the largest lower bound (smallest upper bound) of two elements. Just like graphs, lattices are a fundamental structure that occurs across domains including social data analysis, natural language processing, computational chemistry and biology, and database theory. In this paper we introduce discrete-lattice signal processing (DLSP), an SP framework for data, or signals, indexed by such lattices. We use the meet (or join) to define a shift operation and derive associated notions of filtering, Fourier basis and transform, and frequency response. We show that the spectrum of a lattice signal inherits the lattice structure of the signal domain and derive a sampling theorem. Finally, we show two prototypical applications: spectral analysis of formal concept lattices in social science and sampling and Wiener filtering of multiset lattices in combinatorial auctions. Formal concept lattices are a compressed representation of relations between objects and attributes. Since relations are equivalent to bipartite graphs and hypergraphs, DLSP offers a form of Fourier analysis for these structures. Comment: 13 pages |
Databáze: | arXiv |
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