Wiener Filter on Meet/Join Lattices

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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“…Namely, we first define an energy-preserving shift on which the Wiener filter is built. Part of this subsection are submitted in [42].…”

Section: B Samplingmentioning

confidence: 99%

Discrete Signal Processing on Meet/Join Lattices

Püschel

Seifert

Wendler

2021

IEEE Trans. Signal Process.

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“…Direct predecessors of our work is SP on powersets, i.e., with set functions [24], [25], and SP on meet/join lattices [26], [27], [28]. Both are very special cases of posets and constitute special cases of the framework proposed in this paper.…”

Section: Introductionmentioning

confidence: 99%

Causal Fourier Analysis on Directed Acyclic Graphs and Posets

Seifert¹,

Wendler²,

Püschel³

2022

Preprint

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We present a novel form of Fourier analysis, and associated signal processing concepts, for signals (or data) indexed by edge-weighted directed acyclic graphs (DAGs). This means that our Fourier basis yields an eigendecomposition of a suitable notion of shift and convolution operators that we define. DAGs are the common model to capture causal relationships between data and our framework is causal in that shift, convolution, and Fourier transform are computed only from predecessors in the DAG. The Fourier transform requires the transitive closure of the DAG for which several forms are possible depending on the interpretation of the edge weights. Examples include level of influence, distance, or pollution distribution. Our framework is different from prior GSP: it is specific to DAGs and leverages, and extends, the classical theory of Moebius inversion from combinatorics. For a prototypical application we consider DAGs modeling dynamic networks in which edges change over time. Specifically, we model the spread of an infection on such a DAG obtained from real-world contact tracing data and learn the infection signal from samples assuming sparsity in the Fourier domain.

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