A lattice is a partially ordered set supporting a meet (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 on multiset lattices in combinatorial auctions. Formal concept lattices are a 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.

中文翻译：

---

相遇/连接晶格上的离散信号处理


格是支持相遇（连接）操作的偏序集合，该操作返回两个元素的最大下界（最小上界）。就像图一样，格是跨领域出现的基本结构，包括社会数据分析、自然语言处理、计算化学和生物学以及数据库理论。在本文中，我们介绍了离散格信号处理（DLSP），这是一种由此类格索引的数据或信号的 SP 框架。我们使用满足（或连接）来定义移位操作并导出滤波、傅立叶基和变换以及频率响应的相关概念。我们证明了晶格信号的频谱继承了信号域的晶格结构，并导出了采样定理。最后，我们展示了两个典型应用：社会科学中形式概念格的谱分析以及组合拍卖中多重集格的采样和维纳滤波。形式概念格是对象和属性之间关系的表示。由于关系相当于二部图和超图，因此 DLSP 为这些结构提供了一种傅立叶分析形式。