Multidimensional Persistence Module Classification via Lattice-Theoretic Convolutions

Riess, Hans; Hansen, Jakob

doi:10.48550/arxiv.2011.14057

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Properties and Relation To Gsp1

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2021

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“…Finally, [32] used the lattice convolution from our preliminary [23] to define lattice neural networks.…”

Section: Properties and Relation To Gspmentioning

confidence: 99%

Discrete Signal Processing on Meet/Join Lattices

Püschel

Seifert

Wendler

2021

IEEE Trans. Signal Process.

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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 discretelattice 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.

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“…Finally, [32] used the lattice convolution from our preliminary [23] to define lattice neural networks.…”

Section: Properties and Relation To Gspmentioning

confidence: 99%