Discrete Signal Processing on Meet/Join Lattices

TL;DR

This paper introduces discrete-lattice signal processing (DLSP), a framework for analyzing signals indexed by meet/join lattices, with applications in social science and combinatorial auctions, extending Fourier analysis to complex relational structures.

Contribution

The paper develops a novel DLSP framework that defines shift, filtering, Fourier basis, and sampling on lattice-structured data, bridging signal processing and lattice theory.

Findings

Lattice spectrum inherits the lattice structure of signals.

Derived a sampling theorem for lattice signals.

Demonstrated applications in social science and auction data analysis.

Abstract

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.