交/并格上的离散信号处理
Discrete Signal Processing on Meet/Join Lattices
摘要：
格是一个偏序集，支持一个相遇（或连接）操作，返回两个元素的最大下界（最小上界）。就像图一样，格是一种跨领域的基本结构，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，我们介绍离散格信号处理（DLSP），SP框架的数据，或信号，索引这样的格子。我们使用meet（或join）来定义移位运算，并导出相关的滤波、傅立叶基和变换以及频率响应的概念。我们证明了格信号的频谱继承了信号域的格结构，并导出了一个采样定理。最后，我们展示了两个典型的应用：社会科学中的形式概念格的谱分析和组合拍卖中的多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压缩表示。由于关系等价于二部图和超图，DLSP为这些结构提供了一种傅立叶分析的形式。
英文摘要：
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.
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